A007472 Shifts 2 places left when binomial transform is applied twice with a(0) = a(1) = 1. (Formerly M2812)

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

0,4

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

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

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 *)

Crossrefs

Cf. A351028, A351143.

Row sums of triangle A383235.

Sequence in context: A136628 A151031 A151032 * A292756 A151451 A138938

Adjacent sequences: A007469 A007470 A007471 * A007473 A007474 A007475

Keyword

nonn,nice,eigen

Author

N. J. A. Sloane

Status

approved