A323694 a(n) = [x^n] G(x)^(n+1) / (n+1)^2 for n >= 0, where G(x) is the g.f. of A323693.

1, 1, 6, 80, 1780, 58212, 2609824, 153429696, 11457990000, 1060569950000, 119272908950624, 16028231512305792, 2537651853233579264, 467637448273232473600, 99254069921234336025600, 24041252393859722676492288, 6592065384977916153080977152, 2031547109837445197745286936320, 699189635655553248110260224524800

Offset

0,3

Comments

The g.f. of A323693, G(x), satisfies: [x^n] G(x)^(n+1) = (n+1)^2 * [x^(n-1)] G(x)^(n+1) for n >= 1.

Links

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

Formula

a(n) = [x^(n-1)] G(x)^(n+1) for n >= 1, where G(x) is the g.f. of A323693.

Example

The g.f. of A323693 begins
G(x) = 1 + 2*x + 14*x^2 + 228*x^3 + 6332*x^4 + 255800*x^5 + 13862744*x^6 + 962576816*x^7 + 83146713104*x^8 + 8746885895136*x^9 + ...
The table of coefficients of x^k in G(x)^n starts as
n=1: [1, 2, 14, 228, 6332, 255800, 13862744, ...];
n=2: [1, 4, 32, 512, 13772, 543312, 28977968, ...];
n=3: [1, 6, 54, 860, 22488, 866448, 45462704, ...];
n=4: [1, 8, 80, 1280, 32664, 1229568, 63445984, ...];
n=5: [1, 10, 110, 1780, 44500, 1637512, 83069960, ...];
n=6: [1, 12, 144, 2368, 58212, 2095632, 104491088, ...];
n=7: [1, 14, 182, 3052, 74032, 2609824, 127881376, ...]; ...
RELATED SEQUENCES.
In the above table, the main diagonal begins
[1, 4, 54, 1280, 44500, 2095632, 127881376, 9819500544, ...]
which, when divided by n^2, yields this sequence:
[1, 1, 6, 80, 1780, 58212, 2609824, 153429696, 11457990000, ...]
and also yields the secondary diagonal in the above table.

Prog

(PARI) {a(n) = my(A=[1], V); for(m=2, n+1, A=concat(A, 0); V=Vec(Ser(A)^m); A[#A] = V[#A-1]*m - V[#A]/m ); polcoeff( Ser(A)^(n+1), n)/(n+1)^2}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A323693.

Sequence in context: A349528 A132616 A349657 * A077393 A264694 A358037

Adjacent sequences: A323691 A323692 A323693 * A323695 A323696 A323697

Keyword

nonn

Author

Paul D. Hanna, Feb 20 2019

Status

approved