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A304864 O.g.f. A(x) satisfies: [x^n] exp( n*(n+3) * x ) / A(x) = 0 for n>0. 3
1, 4, 26, 288, 5012, 122608, 3869456, 148838816, 6721823600, 347434618432, 20180665251360, 1299399587904384, 91769604540962816, 7049102617933604352, 584848346900868001792, 52109481481410100183552, 4961586770799906448318208, 502707358017324652042259456, 54000226687663791374322245120, 6129804668943947684749062516736, 733179029209444818691965317379072 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

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

EXAMPLE

O.g.f.: A(x) = 1 + 4*x + 26*x^2 + 288*x^3 + 5012*x^4 + 122608*x^5 + 3869456*x^6 + 148838816*x^7 + 6721823600*x^8 + 347434618432*x^9 + 20180665251360*x^10 + ...

ILLUSTRATION OF DEFINITION.

The table of coefficients of x^k/k! in exp(n*(n+3)*x) / A(x) begins:

n=0: [1, -4, -20, -864, -72576, -10060800, -2068882560, -589017945600, ...];

n=1: [1, 0, -36, -1232, -89088, -11667456, -2328963200, -650497926144, ...];

n=2: [1, 6, 0, -1664, -125136, -14853600, -2803074560, -757869964800, ...];

n=3: [1, 14, 160, 0, -162000, -20768352, -3651775488, -937273259520, ...];

n=4: [1, 24, 540, 10000, 0, -26468352, -5107476608, -1241737082880, ...];

n=5: [1, 36, 1260, 41536, 1133184, 0, -6460818560, -1740188582400, ...];

n=6: [1, 50, 2464, 118368, 5374512, 202760544, 0, -2192436486144, ...];

n=7: [1, 66, 4320, 279136, 17619504, 1054101600, 52553405440, 0, ...];

n=8: [1, 84, 7020, 582400, 47760000, 3832731648, 292170316672, 18603667330560, 0, ...]; ...

RELATED SERIES.

The logarithmic derivative of A(x) yields:

A'(x)/A(x) = 4 + 36*x + 616*x^2 + 15496*x^3 + 504624*x^4 + 19947072*x^5 + 921521248*x^6 + 48536700064*x^7 + 2864002270720*x^8 + 186878075521216*x^9 + ...

1 - 1/A(x) = 4*x + 10*x^2 + 144*x^3 + 3024*x^4 + 83840*x^5 + 2873448*x^6 + 116868640*x^7 + 5488631808*x^8 + 291890096640*x^9 + 17321970359200*x^10 + ...

PROG

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

for(n=0, 25, print1( a(n), ", "))

CROSSREFS

Cf. A304322, A304318, A304319, A304863, A304865.

Sequence in context: A145164 A113078 A177451 * A167811 A156306 A054592

Adjacent sequences:  A304861 A304862 A304863 * A304865 A304866 A304867

KEYWORD

nonn

AUTHOR

Paul D. Hanna, May 19 2018

STATUS

approved

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Last modified September 24 16:57 EDT 2022. Contains 356943 sequences. (Running on oeis4.)