A375440 Expansion of g.f. A(x) satisfying 0 = Sum_{k=0..n} (-1)^k * binomial(2*n, 2*k) * ([x^k] A(x)^n) for n >= 1.
1, 1, 5, 65, 1593, 61953, 3476813, 265517441, 26492540401, 3349218907137, 523572565927509, 99215376614955457, 22415450137196941993, 5953820173628518544385, 1837040977427662958973341, 651657636773935012586716929, 263375512326578915885862469601, 120319850003020550647400856678401
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 5*x^2 + 65*x^3 + 1593*x^4 + 61953*x^5 + 3476813*x^6 + 265517441*x^7 + 26492540401*x^8 + ... RELATED TABLES. The table of coefficients of x^k in A(x)^n begins: n=1: [1, 1, 5, 65, 1593, 61953, 3476813, ...]; n=2: [1, 2, 11, 140, 3341, 127742, 7097687, ...]; n=3: [1, 3, 18, 226, 5259, 197637, 10869476, ...]; n=4: [1, 4, 26, 324, 7363, 271928, 14799444, ...]; n=5: [1, 5, 35, 435, 9670, 350926, 18895290, ...]; n=6: [1, 6, 45, 560, 12198, 434964, 23165174, ...]; ... from which we may illustrate the defining property given by 0 = Sum_{k=0..n} (-1)^k * binomial(2*n, 2*k) * ([x^k] A(x)^n). Using the coefficients in the table above, we see that n=1: 0 = 1*1 - 1*1; n=2: 0 = 1*1 - 6*2 + 1*11; n=3: 0 = 1*1 - 15*3 + 15*18 - 1*226; n=4: 0 = 1*1 - 28*4 + 70*26 - 28*324 + 1*7363; n=5: 0 = 1*1 - 45*5 + 210*35 - 210*435 + 45*9670 - 1*350926; n=6: 0 = 1*1 - 66*6 + 495*45 - 924*560 + 495*12198 - 66*434964 + 1*23165174; ... The triangle A086645(n,k) = binomial(2*n, 2*k) begins: n=0: 1; n=1: 1, 1; n=2: 1, 6, 1; n=3: 1, 15, 15, 1; n=4: 1, 28, 70, 28, 1; n=5: 1, 45, 210, 210, 45, 1; n=6: 1, 66, 495, 924, 495, 66, 1; ...
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..200
Programs
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PARI
{a(n) = my(A=[1],m); for(i=1, n, A=concat(A, 0); m=#A-1; A[m+1] = sum(k=0, m, (-1)^(m-k+1) * binomial(2*m, 2*k) * polcoef(Ser(A)^m, k) )/m ); A[n+1]} for(n=0, 20, print1(a(n), ", "))
Formula
a(n) ~ c * 2^(4*n) * n^(2*n + 1/2) / (Pi^(2*n) * exp(2*n)), where c = 7.23682343848882192289996... - Vaclav Kotesovec, Sep 12 2024
Comments