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A303061 G.f. A(x) satisfies: [x^(n-1)] (1+x)^(n*(n-1)) / A(x)^n = 0 for n>1. 2
1, 1, 1, 7, 87, 1667, 42971, 1387941, 53739797, 2421203261, 124265293581, 7150727869627, 455701200668539, 31846907669892495, 2421141672213472919, 198897819736367366729, 17556316040185549675881, 1656973308228250148662329, 166509657562826568857464281, 17749793745710561363581851663, 2000554650909636157531234301439 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..300

FORMULA

a(n) + a(n-1) = A303060(n) for n>=0.

a(n) ~ c * d^n * n! / n^2, where d = -4 / (LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 6.17655460948348035823168... and c = 0.06049920104... - Vaclav Kotesovec, Oct 06 2020

EXAMPLE

G.f.: A(x) = 1 + x + x^2 + 7*x^3 + 87*x^4 + 1667*x^5 + 42971*x^6 + 1387941*x^7 + 53739797*x^8 + 2421203261*x^9 + 124265293581*x^10 + ...

ILLUSTRATION OF DEFINITION.

The table of coefficients in (1+x)^(n*(n-1)) / A(x)^n begins:

n=1: [1, -1, 0, -6, -74, -1500, -39688, -1302742, ...];

n=2: [1, 0, -2, -12, -159, -3136, -82180, -2680752, ...];

n=3: [1, 3, 0, -26, -300, -5454, -137764, -4398210, ...];

n=4: [1, 8, 24, 0, -548, -9576, -223760, -6847536, ...];

n=5: [1, 15, 100, 350, 0, -16022, -376660, -10771830, ...];

n=6: [1, 24, 270, 1844, 7641, 0, -596908, -17643792, ...];

n=7: [1, 35, 588, 6258, 46186, 224196, 0, -26940146, ...];

n=8: [1, 48, 1120, 16864, 182640, 1478160, 8281968, 0, ...]; ...

in which the main diagonal equals all zeros after the initial term, illustrating that [x^(n-1)] (1+x)^(n*(n-1)) / A(x)^n = 0 for n>1.

PROG

(PARI) {a(n) = my(A=[1]); for(m=1, n+1, A=concat(A, 0); A[m] = Vec( (1+x +x*O(x^n))^(m*(m-1))/Ser(A)^m )[m]/m ); A[n+1]}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A303060.

Sequence in context: A279845 A231447 A020556 * A007803 A034219 A034238

Adjacent sequences:  A303058 A303059 A303060 * A303062 A303063 A303064

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Apr 17 2018

STATUS

approved

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Last modified August 6 00:46 EDT 2021. Contains 346493 sequences. (Running on oeis4.)