A007472 Shifts 2 places left when binomial transform is applied twice with a(0) = a(1) = 1.
1, 1, 1, 3, 9, 29, 105, 431, 1969, 9785, 52145, 296155, 1787385, 11428949, 77124569, 546987143, 4062341601, 31502219889, 254500383457, 2137863653811, 18639586581097, 168387382189709, 1573599537048265, 15189509662516063, 151243491212611217, 1551565158004180137
Offset: 0
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..576
- M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
- M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
- N. J. A. Sloane, Transforms
Programs
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Maple
bintr:= proc(p) local b; b:= proc(n) option remember; add(p(k)*binomial(n,k), k=0..n) end end: b:= (bintr@@2)(a): a:= n-> `if`(n<2, 1, b(n-2)): seq(a(n), n=0..30); # Alois P. Heinz, Oct 18 2012 -
Mathematica
bintr[p_] := Module[{b}, b[n_] := b[n] = Sum [p[k]*Binomial[n, k], {k, 0, n}]; b]; b = a // bintr // bintr; a[n_] := If[n<2, 1, b[n-2]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 27 2014, after Alois P. Heinz *) (* another program *) B[x_] := (BesselK[0, 1] + BesselK[1, 1])*BesselI[0, Exp[x]] + (BesselI[1, 1] - BesselI[0, 1])*BesselK[0, Exp[x]]; a[n_] := SeriesCoefficient[FullSimplify[Series[B[x], {x, 0, n}]],n] n! Table[a[n], {n, 0, 30}] (* Ven Popov, Apr 25 2025 *)
Formula
G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 2*x)) / (1 - 2*x). - Ilya Gutkovskiy, Jan 30 2022
E.g.f.: (BesselK(0, 1) + BesselK(1, 1)) * BesselI(0, exp(x)) + (BesselI(1, 1) - BesselI(0, 1)) * BesselK(0, exp(x)). - Ven Popov, Apr 25 2025