A298694 - OEIS

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A298694

G.f. A(x) satisfies: A(x) = Sum_{n>=0} binomial( n*(n+1), n)/(n+1) * x^n / A(x)^(n^2).

1, 1, 4, 32, 419, 8052, 207784, 6724274, 260396693, 11697865930, 596886780272, 34072732137625, 2151062784054901, 148819021611467291, 11198412956841549966, 910736443741061568539, 79616310026220269203631, 7446056807577515910468813, 741918566779386113373532994, 78467177619239380045368550016, 8779922184077661414128958823323

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..20.

EXAMPLE

G.f.: A(x) = 1 + x + 4*x^2 + 32*x^3 + 419*x^4 + 8052*x^5 + 207784*x^6 + 6724274*x^7 + 260396693*x^8 + 11697865930*x^9 + 596886780272*x^10 + 34072732137625*x^11 + 2151062784054901*x^12 + 148819021611467291*x^13 + 11198412956841549966*x^14 + 910736443741061568539*x^15 + ...

such that

A(x) = 1 + C(2,1)/2*x/A(x) + C(6,2)/3*x^2/A(x)^4 + C(12,3)/4*x^3/A(x)^9 + C(20,4)/5*x^4/A(x)^16 + C(30,5)/6*x^5/A(x)^25 + C(42,6)/7*x^6/A(x)^36 + C(56,7)/8*x^7/A(x)^49 + ...

more explicitly,

A(x) = 1 + x/A(x) + 5*x^2/A(x)^4 + 55*x^3/A(x)^9 + 969*x^4/A(x)^16 + 23751*x^5/A(x)^25 + 749398*x^6/A(x)^36 + 28989675*x^7/A(x)^49 + ... + A135861(n)*x^n/A(x)^(n^2) + ...

MATHEMATICA

terms = 21; A[_] = 1; Do[A[x_] = 1 + Sum[Binomial[n*(n+1), n]/(n+1)*x^n/ A[x]^(n^2), {n, terms}] + O[x]^terms, {terms}]; CoefficientList[A[x], x] (* Jean-François Alcover, Feb 09 2018 *)

PROG

(PARI) {a(n) = my(A=[1]); for(i=1, n, A = Vec(sum(m=0, #A, binomial(m*(m+1), m)/(m+1) * x^m/Ser(A)^(m^2) ))); A[n+1]}

for(n=0, 20, print1(a(n), ", "))

CROSSREFS

Cf. A298691, A298692, A298693, A135861.

Sequence in context: A350465 A229548 A005172 * A222685 A322735 A140178

Adjacent sequences: A298691 A298692 A298693 * A298695 A298696 A298697

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 03 2018

STATUS

approved