A377098 G.f. A(x) = Sum_{n>=0} a(n)*x^n where a(n) = Sum_{k=0..n-1} [x^k] A(x)^(k*(n-k)) for n >= 1 with a(0) = 1.
1, 1, 2, 8, 55, 525, 6202, 85842, 1350421, 23687392, 457238998, 9620344475, 219011293036, 5363006495793, 140567134618434, 3927060955253388, 116510112059820553, 3658928109471912657, 121273249515650581850, 4231012832296844451474, 155003839703746214942229, 5949765253601511005012122
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 55*x^4 + 525*x^5 + 6202*x^6 + 85842*x^7 + 1350421*x^8 + 23687392*x^9 + 457238998*x^10 + 9620344475*x^11 + 219011293036*x^12 + ... By definition, a(n) equals the sum of the coefficients of x^k in A(x)^(k*(n-k)), k = 0..n-1, for n >= 1, as illustrated below. a(1) = [x^0] A(x)^0; a(2) = 1 + [x^1] A(x)^1; a(3) = 1 + [x^1] A(x)^2 + [x^2] A(x)^2; a(4) = 1 + [x^1] A(x)^3 + [x^2] A(x)^4 + [x^3] A(x)^3; a(5) = 1 + [x^1] A(x)^4 + [x^2] A(x)^6 + [x^3] A(x)^6 + [x^4] A(x)^4; a(6) = 1 + [x^1] A(x)^5 + [x^2] A(x)^8 + [x^3] A(x)^9 + [x^4] A(x)^8 + [x^5] A(x)^5; a(7) = 1 + [x^1] A(x)^6 + [x^2] A(x)^10 + [x^3] A(x)^12 + [x^4] A(x)^12 + [x^5] A(x)^10 + [x^6] A(x)^6; a(8) = 1 + [x^1] A(x)^7 + [x^2] A(x)^12 + [x^3] A(x)^15 + [x^4] A(x)^16 + [x^5] A(x)^15 + [x^6] A(x)^12 + [x^7] A(x)^7; ... Explicitly, a(1) = 1 = 1; a(2) = 1 + 1 = 2; a(3) = 1 + 2 + 5 = 8; a(4) = 1 + 3 + 14 + 37 = 55; a(5) = 1 + 4 + 27 + 128 + 365 = 525; a(6) = 1 + 5 + 44 + 300 + 1406 + 4446 = 6202; a(7) = 1 + 6 + 65 + 580 + 3795 + 17892 + 63503 = 85842; a(8) = 1 + 7 + 90 + 995 + 8460 + 53088 + 258212 + 1029568 = 1350421; ... RELATED TABLES. The table of coefficients of x^k in A(x)^n begins as follows. n\k 0 1 2 3 4 5 6 7 A^1 = [1, 1, 2, 8, 55, 525, 6202, 85842, ...]; A^2 = [1, 2, 5, 20, 130, 1192, 13738, 187068, ...]; A^3 = [1, 3, 9, 37, 231, 2037, 22877, 306201, ...]; A^4 = [1, 4, 14, 60, 365, 3104, 33944, 446208, ...]; A^5 = [1, 5, 20, 90, 540, 4446, 47330, 610580, ...]; A^6 = [1, 6, 27, 128, 765, 6126, 63503, 803424, ...]; A^7 = [1, 7, 35, 175, 1050, 8218, 83020, 1029568, ...]; A^8 = [1, 8, 44, 232, 1406, 10808, 106540, ...]; A^9 = [1, 9, 54, 300, 1845, 13995, 134838, ...]; A^10 = [1, 10, 65, 380, 2380, 17892, 168820, ...]; A^11 = [1, 11, 77, 473, 3025, 22627, 209539, ...]; A^12 = [1, 12, 90, 580, 3795, 28344, 258212, ...]; A^13 = [1, 13, 104, 702, 4706, 35204, 316238, ...]; A^14 = [1, 14, 119, 840, 5775, 43386, 385217, ...]; A^15 = [1, 15, 135, 995, 7020, 53088, 466970, ...]; A^16 = [1, 16, 152, 1168, 8460, 64528, 563560, ...]; ...
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..300
Programs
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PARI
{a(n) = my(A=[1]); for(m=1,n, A=concat(A,0); A[#A] = 1 + sum(k=1,m-1,(polcoeff(Ser(A)^(k*(m-k)),k)) )); A[n+1]} for(n=0,30,print1(a(n),", "))
Formula
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) a(n) = Sum_{k=0..n-1} [x^k] A(x)^(k*(n-k)) for n >= 1, with a(0) = 1.
(2) A(x) = 1 + x*Sum_{n>=0} x^n/n! * ( d^n/dy^n A(y)^n/(1 - x*A(y)^n) ) evaluated at y = 0.
a(n) ~ c * n! * n^alpha / LambertW(1)^n, where alpha = 0.68670155428... and c = 0.09981115968806..., conjecture: alpha = 2*LambertW(1) - 3 + 4/(1 + LambertW(1)) = 0.6867015542800108601...- Vaclav Kotesovec, Nov 23 2024
Comments