A300874
O.g.f. A(x) satisfies: [x^n] exp( n*(n+1) * A(x) ) = 2*n * [x^(n-1)] exp( n*(n+1) * A(x) ) for n>=1.
Original entry on oeis.org
1, 1, 6, 78, 1560, 41484, 1361640, 52824144, 2355612192, 118455668960, 6624336466880, 407637626194080, 27374154691010816, 1992569727194556608, 156335075280459423360, 13158244845212096286720, 1183162080050737698802176, 113244610738097834450007552, 11500380596282466998941623296, 1235555832300741998445513374720, 140061215510759508434434106953728
Offset: 1
O.g.f.: A(x) = x + x^2 + 6*x^3 + 78*x^4 + 1560*x^5 + 41484*x^6 + 1361640*x^7 + 52824144*x^8 + 2355612192*x^9 + 118455668960*x^10 + ...
where
exp(A(x)) = 1 + x + 3*x^2/2! + 43*x^3/3! + 2041*x^4/4! + 197721*x^5/5! + 31094251*x^6/6! + 7086479443*x^7/7! + 2187876597873*x^8/8! + 874871971357681*x^9/9! + ... + A300873(n)*x^n/n! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp( n*(n+1) * A(x) ) begins:
n=1: [(1), (2), 4, 52/3, 560/3, 52304/15, 4048864/45, 914958416/315, ...];
n=2: [1, (6), (24), 108, 864, 67104/5, 1601424/5, 348254352/35, ...];
n=3: [1, 12, (84), (504), 3600, 211968/5, 4273776/5, 860107104/35, ...];
n=4: [1, 20, 220, (5560/3), (44480/3), 438400/3, 20480720/9, 3534944800/63, ...];
n=5: [1, 30, 480, 5580, (55440), (554400), 6991920, 947466000/7, ...];
n=6: [1, 42, 924, 14364, 181440, (10403568/5), (124842816/5), 1922103792/5, ...];
n=7: [1, 56, 1624, 98224/3, 1566992/3, 107909312/15, (4208547616/45), (58919666624/45), ...]; ...
in which the coefficients in parenthesis are related by
2 = 2*1*(1); 24 = 2*2*(6); 504 = 2*3*(84); 44480/3 = 2*4*(5560/3); 554400 = 2*5*(55440); 124842816/5 = 2*6*(10403568/5); ...
illustrating: [x^n] exp( n*(n+1) * A(x) ) = 2*n * [x^(n-1)] exp( n*(n+1) * A(x) ).
-
{a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)*(#A))); A[#A] = (2*(#A-1)*V[#A-1] - V[#A])/(#A-1)/(#A) ); polcoeff( log(Ser(A)), n)}
for(n=1, 30, print1(a(n), ", "))
A300615
O.g.f. A(x) satisfies: [x^n] exp( n^5 * A(x) ) = n^5 * [x^(n-1)] exp( n^5 * A(x) ) for n>=1.
Original entry on oeis.org
1, 16, 19683, 142475264, 3436799053125, 212148041589128016, 28458158819417861315152, 7380230750280159370894934016, 3385049575573746853297963891959753, 2561548157856026756893458765378989150000, 3026444829408778969259555715061437179090541565, 5340113530831632053993990154143996936096662034267136
Offset: 1
O.g.f.: A(x) = x + 16*x^2 + 19683*x^3 + 142475264*x^4 + 3436799053125*x^5 + 212148041589128016*x^6 + 28458158819417861315152*x^7 + ...
where
exp(A(x)) = 1 + x + 33*x^2/2! + 118195*x^3/3! + 3419881993*x^4/4! + 412433022394701*x^5/5! + 152749066271797582081*x^6/6! + ... + A300614(n)*x^n/n! + ...
such that: [x^n] exp( n^5 * A(x) ) = n^5 * [x^(n-1)] exp( n^5 * A(x) ).
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{a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^5)); A[#A] = ((#A-1)^5*V[#A-1] - V[#A])/(#A-1)^5 ); polcoeff( log(Ser(A)), n)}
for(n=1, 20, print1(a(n), ", "))
A300625
Table of row functions R(n,x) that satisfy: [x^k] exp( k^n * R(n,x) ) = k^n * [x^(k-1)] exp( k^n * R(n,x) ) for k>=1, n>=1, read by antidiagonals.
Original entry on oeis.org
1, 1, 1, 1, 2, 3, 1, 4, 27, 14, 1, 8, 243, 736, 85, 1, 16, 2187, 40448, 30525, 621, 1, 32, 19683, 2351104, 12519125, 1715454, 5236, 1, 64, 177147, 142475264, 6153518125, 6111917748, 123198985, 49680, 1, 128, 1594323, 8856272896, 3436799053125, 31779658925496, 4308276119854, 10931897664, 521721, 1, 256, 14348907, 558312194048, 2049047412828125, 212148041589128016, 287364845865893467, 4151360558858752, 1172808994833, 5994155
Offset: 1
This table of coefficients T(n,k) begins:
n=1: [1, 1, 3, 14, 85, 621, 5236, 49680, ...];
n=2: [1, 2, 27, 736, 30525, 1715454, 123198985, 10931897664, ...];
n=3: [1, 4, 243, 40448, 12519125, 6111917748, 4308276119854, ..];
n=4: [1, 8, 2187, 2351104, 6153518125, 31779658925496, ...];
n=5: [1, 16, 19683, 142475264, 3436799053125, 212148041589128016, ...];
n=6: [1, 32, 177147, 8856272896, 2049047412828125, 1569837215111038900704, ...];
n=7: [1, 64, 1594323, 558312194048, 1256793474918203125, 12020665333382306853887808, ...]; ...
such that row functions R(n,x) = Sum_{k>=1} T(n,k)*x^k satisfy:
[x^k] exp( k^n * R(n,x) ) = k^n * [x^(k-1)] exp( k^n * R(n,x) ) for k>=1.
Row functions R(n,x) begin:
R(1,x) = x + x^2 + 3*x^3 + 14*x^4 + 85*x^5 + 621*x^6 + 5236*x^7 + 49680*x^8 + ...
R(2,x) = x + 2*x^2 + 27*x^3 + 736*x^4 + 30525*x^5 + 1715454*x^6 + ...
R(3,x) = x + 4*x^2 + 243*x^3 + 40448*x^4 + 12519125*x^5 + 6111917748*x^6 + ...
R(4,x) = x + 8*x^2 + 2187*x^3 + 2351104*x^4 + 6153518125*x^5 + ...
etc.
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{T(n, k) = my(A=[1]); for(i=1, k+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^n)); A[#A] = ((#A-1)^n * V[#A-1] - V[#A])/(#A-1)^n ); polcoeff( log(Ser(A)), k)}
/* Print as a table of row functions: */
for(n=1, 8, for(k=1, 8, print1(T(n, k), ", ")); print(""))
/* Print as a flattened triangle: */
for(n=1, 12, for(k=1, n-1, print1(T(n-k, k), ", ")); )
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