A325452 G.f. A(x) satisfies: Sum_{n>=0} x^n * A(x)^(n*(n+1)/2) = Sum_{n>=0} x^n / [Product_{k=1..n} 1 - k*x].

1, 1, 1, 2, 7, 31, 157, 873, 5213, 32994, 219529, 1527181, 11066057, 83297249, 650046972, 5251478949, 43865544600, 378477776926, 3370208561296, 30947767628626, 292841013687401, 2853294268280367, 28606103960232380, 294886674017837164, 3123374417314899189, 33966912560799919837, 379005891541426104696, 4336049700830955390152, 50828681335405503653495, 610107351974853060787970

Offset

0,4

Links

Table of n, a(n) for n=0..29.

Example

G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 14*x^4 + 81*x^5 + 554*x^6 + 4175*x^7 + 33894*x^8 + 292482*x^9 + 2658803*x^10 + 25312031*x^11 + 251337905*x^12 + ...
such that the following series are equal
B(x) = 1 + x*A(x) + x^2*A(x)^3 + x^3*A(x)^6 + x^4*A(x)^10 + x^5*A(x)^15 + x^6*A(x)^21 + x^7*A(x)^28 + x^8*A(x)^36 + x^9*A(x)^45 + ...
B(x) = 1 + x/(1-x) + x^2/((1-x)*(1-2*x)) + x^3/((1-x)*(1-2*x)*(1-3*x)) + x^4/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)) + x^5/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)) + x^6/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)*(1-6*x)) + ....
where B(x) is the o.g.f. of the Bell numbers:
B(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 52*x^5 + 203*x^6 + 877*x^7 + 4140*x^8 + 21147*x^9 + 115975*x^10 + ... + A000110(n)*x^n + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff( sum(m=0, #A, x^m*( 1/prod(k=1, m, 1-k*x +x*O(x^#A)) - Ser(A)^(m*(m+1)/2)) ), #A) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A000110, A325453.

Sequence in context: A030873 A030913 A181066 * A030945 A088554 A107595

Adjacent sequences: A325449 A325450 A325451 * A325453 A325454 A325455

Keyword

nonn

Author

Paul D. Hanna, Apr 30 2019

Status

approved