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A027826 Inverse binomial transform of a_0 = 1, a_1, a_2, etc. is a_0, 0, a_1, 0, a_2, 0, etc. 7
1, 1, 2, 4, 9, 21, 50, 120, 290, 706, 1732, 4280, 10644, 26612, 66824, 168384, 425481, 1077529, 2733746, 6945812, 17669149, 44994345, 114682042, 292544200, 746831570, 1907983346, 4877966628, 12479883736, 31951158024, 81858610968, 209865391600, 538408691456 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The self-convolution equals A051163. - Paul D. Hanna, Nov 23 2004

Equals row sums of triangle A152193. - Gary W. Adamson, Nov 28 2008

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

N. J. A. Sloane, Transforms

FORMULA

G.f. A(x) satisfies A(x^2) = A(x/(1+x))/(1+x) and A(x) = A(x^2/(1-x)^2)/(1-x).

The recursive formula A[n+1] = A[n](x^2/(1-x)^2)/(1-x), A[0]=1, yields exactly 2^n terms after n iterations: A(x) - A[n](x) = x^(2^n) + (2^n+1)*x^(2^n+1) + O(x^(2^n+2)). For example, A[4] = (1-x)^3*(1-2*x-x^2)/((1-2*x)(1-4*x+4*x^2-2*x^4)) = A(x) - x^16 - 17*x^17 + O(x^18). - M. F. Hasler, Aug 19 2015

EXAMPLE

Array of successive differences (col. 1 is the inverse binomial transform):

1, 1,  2,  4,  9, 21, 50, ...

0, 1,  2,  5, 12, 29, 70, ...

1, 1,  3,  7, 17, 41, ...

0, 2,  4, 10, 24, 59, ...

2, 2,  6, 14, 35, 87, ...

0, 4,  8, 21, 52, ...

4, 4, 13, 31, 79, ...

0, 9, 18, 48, ...

9, 9, 30, ...

...

MAPLE

a:= proc(n) option remember; add(`if`(k=0, 1,

      `if`(k::odd, 0, a(k/2)))*binomial(n, k), k=0..n)

    end:

seq(a(n), n=0..40);  # Alois P. Heinz, Jul 08 2015

MATHEMATICA

a[n_] := a[n] = Sum[If[k == 0, 1, If[OddQ[k], 0, a[k/2]]]*Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 0, 40}] (* Jean-Fran├žois Alcover, Jan 20 2017, translated from Maple *)

PROG

(PARI) a(n)=local(A, m); if(n<0, 0, m=1; A=1+O(x); while(m<=n, m*=2; A=subst(A, x, (x/(1-x))^2)/(1-x)); polcoeff(A, n))

(PARI) a=List(); for(n=1, 100, listput(a, sum(i=1, n\2, a[i]*binomial(n, 2*i), 1))) \\ M. F. Hasler, Aug 19 2015

CROSSREFS

Cf. A051163.

Cf. A152193. - Gary W. Adamson, Nov 28 2008

Sequence in context: A024537 A171842 A296201 * A261664 A091964 A092423

Adjacent sequences:  A027823 A027824 A027825 * A027827 A027828 A027829

KEYWORD

nonn,eigen

AUTHOR

Allan C. Wechsler

EXTENSIONS

Incorrect g.f. and formulas removed by R. J. Mathar, Oct 02 2012

Incorrect g.f.s removed by Peter Bala, Jul 07 2015

STATUS

approved

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Last modified March 22 10:07 EDT 2019. Contains 321421 sequences. (Running on oeis4.)