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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-François 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 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.)