Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
generalized Schröder numbers; coordination numbers; crystal ball numbers; stretched Riordan array; triangular matrix; sequence transformation; inversion; left-inverse
generalized Schröder numbers; coordination numbers; crystal ball numbers; stretched Riordan array; triangular matrix; sequence transformation; inversion; left-inverse
Summary:
Infinite lower triangular matrices of generalized Schröder numbers are used to construct a two-parameter class of invertible sequence transformations. Their inverses are given by triangular matrices of coordination numbers. The two-parameter class of Schröder transformations is merged into a one-parameter class of stretched Riordan arrays, the left-inverses of which consist of matrices of crystal ball numbers. Schröder and inverse Schröder transforms of important sequences are calculated.
References:
[1] Aigner M.: Diskrete Mathematik. Vieweg, Braunschweig, 1993. MR 1243412 | Zbl 1109.05001
[2] Bernstein M., Sloane N.J.A.: Some canocial sequences of integers. Linear Algebra Appl. 226–228, (1995), 57–72. MR 1344554
[3] Bower C.G.: \rm transforms2. http://oeis.org/transforms2.html as from August 2011, published electronically.
[4] Conway J.H., Sloane N.J.A.: Low Dimensional lattices VII: Coordination sequences. Proc. Royal Soc. London Ser. A Math. Phys. Eng. Sci. 453 (1997), 2369–2389; {citeseer.ist.psu.edu/article/conway96lowdimensional.html}. DOI 10.1098/rspa.1997.0126 | MR 1480120 | Zbl 1066.11505
[5] Corsani C., Merlini D., Sprugnoli R.: Left-inversion of combinatorial sums. Discrete Math. 180 (1998), 107–122. DOI 10.1016/S0012-365X(97)00110-6 | MR 1603705 | Zbl 0903.05005
[6] Gould H.W., Hsu L.C.: Some new inverse series relations. Duke Math. J. 40 (1973), 885–891. DOI 10.1215/S0012-7094-73-04082-9 | MR 0337652
[7] Krattenthaler C.: A new matrix inverse. Proc. Amer. Math. Soc. 124 (1996), 47–59. DOI 10.1090/S0002-9939-96-03042-0 | MR 1291781 | Zbl 0843.15005
[8] Riordan J.: Combinatorial Identities. Wiley, New York, 1968. MR 0231725 | Zbl 0517.05006
[9] Riordan J.: Inverse relations and combinatorial identities. Amer. Math. Monthly 71 (1964), 485–498. DOI 10.2307/2312584 | MR 0169791 | Zbl 0128.01603
[10] Schröder E.: Vier combinatorische Probleme. Z. Math. Phys. 15 (1870), 361–376.
[11] Schröder J.: Generalized Schröder numbers and the rotation principle. J. Integer Seq. 10 (2007), 1–15, Article 07.7.7, http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Schroder/schroder45.pdf MR 2346050
[12] Sloane N.J.A.: transforms. http://oeis.org/transforms.html as from August 2011, published electronically.
[13] Sloane N.J.A., and Mathematical Community: The On-Line Encyclopedia of Integer Sequences. http://www.research.att.com/ njas/sequences/.
[14] Sprugnoli R.: Riordan arrays and combinatorial sums. Discrete Math. 132 (1994), 267–290. DOI 10.1016/0012-365X(92)00570-H | MR 1297386 | Zbl 0814.05003
[15] Stanley R.P.: Enumerative Combinatorics. Vol. 2, Cambridge University Press, Cambridge, 1999. MR 1676282 | Zbl 0978.05002
[16] Sulanke R.A.: A recurrence restricted by a diagonal condition: generalized Catalan arrays. Fibonacci Q. 27 (1989), 33–46. MR 0981063 | Zbl 0666.10008
Search
Partner of
