A393760 Rectangular array, read by ascending antidiagonals, where the n-th row o.g.f. R(n,x) satisfies: [x^m] exp( m^2*(m + n)*x - m*R(n,x) ) = 0 for m >= 1, n >= 0.

1, 2, 9, 3, 16, 552, 4, 25, 1020, 85842, 5, 36, 1692, 165536, 24653700, 6, 49, 2604, 289538, 49320160, 11219022936, 7, 64, 3792, 471456, 90173820, 23135212080, 7393496092416, 8, 81, 5292, 726674, 153941040, 43882422792, 15630527512432, 6649411839351120, 9, 100, 7140, 1072352, 248954740, 77949764208, 30567719526512, 14347969660399360, 7822998961379912592

Offset

0,2

Comments

Let F(n,x) satisfy [x^m] exp( m*(m+n)*x - m*F(n,x) ) = 0 for m >= 1, then F(n,x) = n*x + G(x) where G(x) is the o.g.f. for A317344; compare with the condition satisfied by the row o.g.f.s of this array.

In general, for fixed r, if g.f. A(x) satisfies: [x^n] exp( n^2*(n+r)*x - n*A(x) ) = 0 for n >= 1, then a(n) ~ sqrt(1-w) * 3^(3*n - 1 + r/3) * n^(2*n - 3/2) / (sqrt(2*Pi) * exp(2*n) * (3-w)^(2*n-1) * w^(n + r/3)), where w = -A226750 = -LambertW(-3*exp(-3)). - Vaclav Kotesovec, Mar 02 2026

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1325 (rows 0..50)

Example

This rectangle of coefficients T(n,k) of x^k in n-th row o.g.f. R(n,x) for n >= 0, k >= 1, begins
  n = 0: [1,   9,  552,   85842,  24653700, ...];
  n = 1: [2,  16, 1020,  165536,  49320160, ...];
  n = 2: [3,  25, 1692,  289538,  90173820, ...];
  n = 3: [4,  36, 2604,  471456, 153941040, ...];
  n = 4: [5,  49, 3792,  726674, 248954740, ...];
  n = 5: [6,  64, 5292, 1072352, 385322400, ...];
  n = 6: [7,  81, 7140, 1527426, 575094060, ...];
  n = 7: [8, 100, 9372, 2112608, 832430320, ...];
  ...
As a triangle,
  n = 0: [1];
  n = 1: [2,  9];
  n = 2: [3, 16,  552];
  n = 3: [4, 25, 1020,  85842];
  n = 4: [5, 36, 1692, 165536,  24653700];
  n = 5: [6, 49, 2604, 289538,  49320160, 11219022936];
  n = 6: [7, 64, 3792, 471456,  90173820, 23135212080,  7393496092416];
  n = 7: [8, 81, 5292, 726674, 153941040, 43882422792, 15630527512432, 6649411839351120];
  ...
Row functions R(n,x) begin
R(0,x) = x + 9*x^2 + 552*x^3 + 85842*x^4 + 24653700*x^5 + ... + A317347(n)*x^n + ...
R(1,x) = 2*x + 16*x^2 + 1020*x^3 + 165536*x^4 + 49320160*x^5 + ... + A393761(n)*x^n + ...
R(2,x) = 3*x + 25*x^2 + 1692*x^3 + 289538*x^4 + 90173820*x^5 + ... + A393762(n)*x^n + ...
R(3,x) = 4*x + 36*x^2 + 2604*x^3 + 471456*x^4 + 153941040*x^5 + ... + A393763(n)*x^n + ...
R(4,x) = 5*x + 49*x^2 + 3792*x^3 + 726674*x^4 + 248954740*x^5 + ... + A393764(n)*x^n + ...
R(5,x) = 6*x + 64*x^2 + 5292*x^3 + 1072352*x^4 + 385322400*x^5 + ... + A393765(n)*x^n + ...
R(6,x) = 7*x + 81*x^2 + 7140*x^3 + 1527426*x^4 + 575094060*x^5 + ... + A393766(n)*x^n + ...
R(7,x) = 8*x + 100*x^2 + 9372*x^3 + 2112608*x^4 + 832430320*x^5 + ... + A393767(n)*x^n + ...
...
where [x^m] exp( m^2*(m + n)*x - m*R(n,x) ) = 0 for m >= 1, n >= 0.

Prog

(PARI) {T(n, k) = my(A=[0], m); for(i=1, k+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^2*(m + n)*x - m*Ser(A)) )[m+1]/m ); polcoef( Ser(A), k)}
\\ Print as a rectangle
for(n=0, 9, for(k=1, 9, print1( T(n, k), ", ")); print(""))
\\ Print as a triangle
for(n=0, 9, for(k=1, n+1, print1( T(n-k+1, k), ", ")); print(""))

Crossrefs

Cf. A317347 (n=0), A393761 (n=1), A393762 (n=2), A393763 (n=3).

Cf. A393764 (n=4), A393765 (n=5), A393766 (n=6), A393767 (n=7).

Cf. A393768 (diagonal).

Sequence in context: A379293 A171534 A163907 * A237860 A088614 A162615

Adjacent sequences: A393757 A393758 A393759 * A393761 A393762 A393763

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 01 2026

Status

approved