A128570 Rectangular table, read by antidiagonals, where the g.f. of row n, R(x,n), satisfies: R(x,n) = 1 + (n+1)*x*R(x,n+1)^2 for n>=0.

1, 1, 1, 1, 2, 4, 1, 3, 12, 28, 1, 4, 24, 114, 276, 1, 5, 40, 288, 1440, 3480, 1, 6, 60, 580, 4440, 22368, 53232, 1, 7, 84, 1020, 10560, 82080, 409248, 955524, 1, 8, 112, 1638, 21420, 226560, 1752000, 8585088, 19672320, 1, 9, 144, 2464, 38976, 523320, 5532960, 42178800, 202733760, 456803328, 1, 10, 180, 3528, 65520, 1068480, 14399280, 150570240, 1127335680, 5317663680, 11810032896, 1, 11, 220, 4860, 103680, 1991808, 32716992, 437433780, 4501422240, 33073099200, 153345634560, 336463895808

Offset: 0

Author

Paul D. Hanna, Mar 11 2007

Keywords

• nonn
• tabl

Comments

Row r > 0 is asymptotic to 2^(2*r) * n^r * A128318(n) / (3^r * r!). - Vaclav Kotesovec, Mar 19 2016

Examples

			Row g.f.s satisfy: R(x,0) = 1 + x*R(x,1)^2, R(x,1) = 1 + 2x*R(x,2)^2,
R(x,2) = 1 + 3x*R(x,3)^2, R(x,3) = 1 + 4x*R(x,4)^2, ...
where the initial rows begin:
R(x,0):[1,1,4,28,276,3480,53232,955524,19672320,456803328,...];
R(x,1):[1,2,12,114,1440,22368,409248,8585088,202733760,...];
R(x,2):[1,3,24,288,4440,82080,1752000,42178800,1127335680,...];
R(x,3):[1,4,40,580,10560,226560,5532960,150570240,4501422240,...];
R(x,4):[1,5,60,1020,21420,523320,14399280,437433780,14479664640,...];
R(x,5):[1,6,84,1638,38976,1068480,32716992,1098069504,39896236800,...];
R(x,6):[1,7,112,2464,65520,1991808,67189248,2469837888,97765355520,..];
R(x,7):[1,8,144,3528,103680,3461760,127569600,5098406400,...];
R(x,8):[1,9,180,4860,156420,5690520,227470320,9821970180,...];
R(x,9):[1,10,220,6490,227040,8939040,385265760,17875608960,..].
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..527

Crossrefs

Rows: A128318, A128571, A128572, A128573, A128574, A128575, A128576; A128577 (square of row 0), A128578 (main diagonal), A128579 (antidiagonal sums).

Cf A268652.

Programs

PARI

{T(n,k)=local(A=1+(n+k+1)*x); for(j=0,k,A=1+(n+k+1-j)*x*A^2 +x*O(x^k));polcoeff(A,k)}
for(n=0, 12, for(k=0, 10, print1(T(n, k), ", ")); print(""))