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A027826
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Inverse binomial transform of a_0 = 1, a_1, a_2, etc. is a_0, 0, a_1, 0, a_2, 0, etc.
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11
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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
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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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
E.g.f.: exp(x) * Sum_{n>=0} a(n) * x^(2*n) / (2*n)!. - Ilya Gutkovskiy, Feb 26 2022
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
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EXAMPLE
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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, ...
...
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MAPLE
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a:= proc(n) option remember; add(`if`(k=0, 1,
`if`(k::odd, 0, a(k/2)))*binomial(n, k), k=0..n)
end:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,eigen
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AUTHOR
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EXTENSIONS
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Incorrect g.f. and formulas removed by R. J. Mathar, Oct 02 2012
Incorrect g.f.s removed by Peter Bala, Jul 07 2015
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STATUS
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approved
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