A230253 Coefficients of power series A(x) such that coefficient of x^n in A(x)^(n+1) equals (n+1)*(n+2)!/2 for n>=0.

1, 3, 3, 6, 27, 162, 1206, 10476, 103059, 1125738, 13473378, 174997908, 2448791838, 36706645908, 586646510796, 9957100024152, 178868488496643, 3390603439026618, 67639341903290730, 1416612563019545220, 31079692422132040170, 712855563504590236860, 17061654943814209044660

Offset: 0

Author

Paul D. Hanna, Oct 13 2013

Keywords

• nonn

Examples

			G.f. A(x) = 1 + 3*x + 3*x^2 + 6*x^3 + 27*x^4 + 162*x^5 + 1206*x^6 +...
The coefficients in A(x)^n begin:
n=1: [(1),3,   3,    6,    27,    162,   1206,   10476,   103059, ...];
n=2: [1, (6), 15,   30,    99,    522,   3582,   29484,   278883, ...];
n=3: [1,  9, (36),  99,   297,   1323,   8208,   63342,   572751, ...];
n=4: [1, 12,  66, (240),  783,   3132,  17298,  123552,  1060155, ...];
n=5: [1, 15, 105,  480, (1800),  7083,  35415,  231660,  1869885, ...];
n=6: [1, 18, 153,  846,  3672, (15120), 71415,  428490,  3226797, ...];
n=7: [1, 21, 210, 1365,  6804,  30240,(141120), 789939,  5529573, ...];
n=8: [1, 24, 276, 2064, 11682,  56736, 270720,(1451520), 9485343, ...];
n=9: [1, 27, 351, 2970, 18873, 100440, 500904, 2643840,(16329600), ...]; ...
where the coefficient of x^n in A(x)^(n+1) equals (n+1)*(n+2)!/2 for n>=0.
Notice that the diagonal above the main diagonal forms A111546.
		

Crossrefs

Cf. A075834, A090753, A111546.

Programs

PARI

{a(n)=polcoeff(x/serreverse(sum(m=1, n+1, (m+1)!/2*x^m)+x^2*O(x^n)), n)}

Formula

[x^n] A(x)^n = A111546(n) for n>=0.