A300619 -id:A300619 - cp's OEIS frontend

A300618 E.g.f. A(x) satisfies: [x^n] A(x)^n = n^3 * [x^(n-1)] A(x)^n for n>=1.

1, 1, 15, 1285, 347065, 224232501, 296201195791, 719274160258585, 2967337954539761265, 19563048191912257746505, 196302561889372679184550831, 2881342883089548932078551914861, 59862434550069057805236434063104105, 1712289828911477479390772271103153886845

Offset: 0

Paul D. Hanna, Mar 10 2018

nonn

Compare e.g.f. to: [x^n] exp(x)^n = [x^(n-1)] exp(x)^n for n>=1.

			E.g.f.: A(x) = 1 + x + 15*x^2/2! + 1285*x^3/3! + 347065*x^4/4! + 224232501*x^5/5! + 296201195791*x^6/6! + 719274160258585*x^7/7! + 2967337954539761265*x^8/8! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^n in A(x)^n begins:
n=1: [(1), (1), 15/2, 1285/6, 347065/24, 74744167/40, ...];
n=2: [1, (2), (16), 1330/3, 88220/3, 56540144/15, ...];
n=3: [1, 3, (51/2), (1377/2), 358875/8, 228121101/40, ...];
n=4: [1, 4, 36, (2852/3), (182528/3), 38352496/5, ...];
n=5: [1, 5, 95/2, 7385/6, (1857145/24), (232143125/24), ...];
n=6: [1, 6, 60, 1530, 94500, (58551624/5), (12647150784/5), ...]; ...
in which the coefficients in parenthesis are related by
1 = 1*1; 16 = 2^3*2; 1377/2 = 3^3*51/2; 182528/3 = 4^3*2852/3; ...
illustrating that: [x^n] A(x)^n = n^3 * [x^(n-1)] A(x)^n.
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + 7*x^2 + 207*x^3 + 14226*x^4 + 1852800*x^5 + 409408077*x^6 + 142286748933*x^7 + 73448832515952*x^8 + ... + A300619(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A182962, A296170, A300590, A300592, A300594, A300596, A300614, A300616, A300619.

{a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(#A-1)); A[#A] = ((#A-1)^3*V[#A-1] - V[#A])/(#A-1) ); n!*A[n+1]}
for(n=0, 20, print1(a(n), ", "))

E.g.f. A(x) satisfies: log(A(x)) = Sum_{n>=1} A300619(n)*x^n, a power series in x with integer coefficients.

A300620 Table of row functions R(n,x) that satisfy: [x^k] exp( k * R(n,x) ) = k^n * [x^(k-1)] exp( k * R(n,x) ) for k>=1, n>=1, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 7, 30, 14, 1, 15, 207, 550, 85, 1, 31, 1290, 14226, 15375, 621, 1, 63, 7803, 340550, 1852800, 601398, 5236, 1, 127, 46830, 8086594, 215528250, 409408077, 31299268, 49680, 1, 255, 280647, 192663030, 25359510515, 280823532696, 142286748933, 2093655600, 521721, 1, 511, 1682130, 4605331346, 3013207159725, 197431364485587, 676005054191880, 73448832515952, 175312873125, 5994155

Offset: 1

Paul D. Hanna, Mar 12 2018

			This table of coefficients T(n,k) begins:
n=1: [1, 1, 3, 14, 85, 621, 5236, ...];
n=2: [1, 3, 30, 550, 15375, 601398, 31299268, ...];
n=3: [1, 7, 207, 14226, 1852800, 409408077, 142286748933, ...];
n=4: [1, 15, 1290, 340550, 215528250, 280823532696, 676005054191880, ...];
n=5: [1, 31, 7803, 8086594, 25359510515, 197431364485587, ...];
n=6: [1, 63, 46830, 192663030, 3013207159725, 140620832995924134, ...];
n=7: [1, 127, 280647, 4605331346, 359881205186350, 100749338488125315273, 82972785219971584775198767, ...]; ...
such that row functions R(n,x) = Sum_{k>=1} T(n,k)*x^k satisfy:
[x^k] exp( k * R(n,x) ) = k^n * [x^(k-1)] exp( k * 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 + 3*x^2 + 30*x^3 + 550*x^4 + 15375*x^5 + 601398*x^6 + 31299268*x^7 + ...
R(3,x) = x + 7*x^2 + 207*x^3 + 14226*x^4 + 1852800*x^5 + 409408077*x^6 + ...
R(4,x) = x + 15*x^2 + 1290*x^3 + 340550*x^4 + 215528250*x^5 + 280823532696*x^6 + ...
etc.

Paul D. Hanna, Table of n, a(n) for n = 1..435 of rows 1..30 as a flattened table read by antidiagonals.

Cf. A088716 (row 1), A300617 (row 2), A300619 (row 3).

{T(n,k) = my(A=[1]); for(i=1, k+1, A=concat(A, 0); V=Vec(Ser(A)^(#A-1)); A[#A] = ((#A-1)^n*V[#A-1] - V[#A])/(#A-1) ); polcoeff( log(Ser(A)), k)}
for(n=1, 8, for(k=1,8, print1(T(n,k), ", "));print(""))

cp's OEIS Frontend

A300618 E.g.f. A(x) satisfies: [x^n] A(x)^n = n^3 * [x^(n-1)] A(x)^n for n>=1.

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