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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.

%I A027826 #49 Jan 15 2023 09:51:47
%S A027826 1,1,2,4,9,21,50,120,290,706,1732,4280,10644,26612,66824,168384,
%T A027826 425481,1077529,2733746,6945812,17669149,44994345,114682042,292544200,
%U A027826 746831570,1907983346,4877966628,12479883736,31951158024,81858610968,209865391600,538408691456
%N A027826 Inverse binomial transform of a_0 = 1, a_1, a_2, etc. is a_0, 0, a_1, 0, a_2, 0, etc.
%C A027826 The self-convolution equals A051163. - _Paul D. Hanna_, Nov 23 2004
%C A027826 Equals row sums of triangle A152193. - _Gary W. Adamson_, Nov 28 2008
%H A027826 Alois P. Heinz, <a href="/A027826/b027826.txt">Table of n, a(n) for n = 0..1000</a>
%H A027826 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A027826 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).
%F A027826 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
%F A027826 E.g.f.: exp(x) * Sum_{n>=0} a(n) * x^(2*n) / (2*n)!. - _Ilya Gutkovskiy_, Feb 26 2022
%F A027826 The expansion of exp(Sum_{n >= 1} a(n)*(2*x)^n/n!) = 1 + 2*x + 6*x^2 + 20*x^3 + 74*x^4 + 292*x^5 + 1204*x^6 + ... appears to have integer coefficients. Equivalently, the Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for positive integers k and n and all primes p >= 3. - _Peter Bala_, Jan 11 2023
%e A027826 Array of successive differences (col. 1 is the inverse binomial transform):
%e A027826 1, 1,  2,  4,  9, 21, 50, ...
%e A027826 0, 1,  2,  5, 12, 29, 70, ...
%e A027826 1, 1,  3,  7, 17, 41, ...
%e A027826 0, 2,  4, 10, 24, 59, ...
%e A027826 2, 2,  6, 14, 35, 87, ...
%e A027826 0, 4,  8, 21, 52, ...
%e A027826 4, 4, 13, 31, 79, ...
%e A027826 0, 9, 18, 48, ...
%e A027826 9, 9, 30, ...
%e A027826 ...
%p A027826 a:= proc(n) option remember; add(`if`(k=0, 1,
%p A027826       `if`(k::odd, 0, a(k/2)))*binomial(n, k), k=0..n)
%p A027826     end:
%p A027826 seq(a(n), n=0..40);  # _Alois P. Heinz_, Jul 08 2015
%t A027826 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 *)
%o A027826 (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))
%o A027826 (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
%Y A027826 Cf. A051163.
%Y A027826 Cf. A152193. - _Gary W. Adamson_, Nov 28 2008
%K A027826 nonn,eigen
%O A027826 0,3
%A A027826 _Allan C. Wechsler_
%E A027826 Incorrect g.f. and formulas removed by _R. J. Mathar_, Oct 02 2012
%E A027826 Incorrect g.f.s removed by _Peter Bala_, Jul 07 2015