A007472
Shifts 2 places left when binomial transform is applied twice with a(0) = a(1) = 1.
Original entry on oeis.org
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
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- 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
-
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
-
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 *)
A351053
G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 3*x)) / (1 - 3*x).
Original entry on oeis.org
0, 1, 0, 1, 6, 28, 126, 613, 3438, 22159, 157362, 1189126, 9436320, 78690781, 692478684, 6439539457, 63106488618, 648453907216, 6952719052134, 77521908188737, 897132401326458, 10764085132255807, 133774484448519294, 1720018195807299418, 22847325911461934352
Offset: 0
-
nmax = 24; A[] = 0; Do[A[x] = x + x^2 A[x/(1 - 3 x)]/(1 - 3 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 0; a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] 3^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 24}]
A351128
G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 4*x)) / (1 - 4*x).
Original entry on oeis.org
0, 1, 0, 1, 8, 49, 280, 1649, 10800, 81505, 696400, 6472033, 63562872, 652984977, 7026210728, 79547049681, 949709767904, 11936248012993, 157219119485216, 2159448120457409, 30811324011852136, 455635009201780977, 6975424580445456056, 110478282815356437809
Offset: 0
-
nmax = 23; A[] = 0; Do[A[x] = x + x^2 A[x/(1 - 4 x)]/(1 - 4 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 0; a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] 4^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 23}]
A351132
G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 5*x)) / (1 - 5*x).
Original entry on oeis.org
0, 1, 0, 1, 10, 76, 530, 3701, 27810, 237151, 2316350, 25135126, 292106400, 3559029501, 45211131460, 600619791201, 8384107777030, 123237338584576, 1904128564485610, 30789744821412401, 518479182191232950, 9057086806410632751, 163745788914416588050
Offset: 0
-
nmax = 22; A[] = 0; Do[A[x] = x + x^2 A[x/(1 - 5 x)]/(1 - 5 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 0; a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] 5^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 22}]
A351161
G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 6*x)) / (1 - 6*x).
Original entry on oeis.org
0, 1, 0, 1, 12, 109, 900, 7309, 62280, 590185, 6402360, 78347593, 1042633908, 14648616757, 214421295132, 3266839420021, 52041902492496, 870810496011793, 15326196662766384, 283049655668743249, 5460180803581446684, 109489002283248831037, 2273856664328893182324
Offset: 0
-
nmax = 22; A[] = 0; Do[A[x] = x + x^2 A[x/(1 - 6 x)]/(1 - 6 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 0; a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] 6^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 22}]
Showing 1-5 of 5 results.
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