A393768 Main diagonal of rectangular array A393760.

1, 16, 1692, 471456, 248954740, 211762028976, 263602136467152, 450793834408701952, 1012717008332245086480, 2890413726695794368777600, 10212281910646409194031226752, 43748238038977256648651851984896, 223402441704169342933506999842426304, 1340718887279590068185812449175232353024

Offset

0,2

Comments

The n-th row o.g.f. R(n,x) of rectangle A393760 satisfies: [x^m] exp( m^2*(m + n)*x - m*R(n,x) ) = 0 for m >= 1, n >= 0. This sequence is the main diagonal a(n) = A393760(n, n+1) for n >= 0.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..201

Formula

a(n) ~ c * d^n * n^(2*n + 1/2) / exp(2*n), where d = 36.47634888601519724695... and c = 4.4928520452446669... - Vaclav Kotesovec, Mar 02 2026

Example

Rectangular array A393760 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, ...];
  ...
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.
The main diagonal in the above table forms this sequence.

Prog

(PARI) {A393760(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)}
for(n=0, 15, print1(A393760(n, n+1), ", "))

Crossrefs

Cf. A393760.

Sequence in context: A263387 A264199 A160306 * A167000 A075413 A265213

Adjacent sequences: A393765 A393766 A393767 * A393769 A393770 A393771

Keyword

nonn

Author

Paul D. Hanna, Mar 01 2026

Status

approved