A322891 Triangle, read by rows, each row n being defined by g.f. Product_{k=1..n} (k + x + 2*k*x^2), for n >= 0.

1, 1, 1, 2, 2, 3, 9, 6, 8, 6, 11, 42, 45, 84, 44, 48, 24, 50, 227, 310, 717, 620, 908, 400, 384, 120, 274, 1425, 2277, 6165, 6917, 12330, 9108, 11400, 4384, 3840, 720, 1764, 10264, 18375, 56367, 74991, 154877, 149982, 225468, 147000, 164224, 56448, 46080, 5040, 13068, 83692, 163585, 556640, 838554, 1948268, 2254625, 3896536, 3354216, 4453120, 2617360, 2678144, 836352, 645120, 40320, 109584, 763244, 1601460, 5955777, 9882432, 25330938, 33402132, 64599201, 66804264, 101323752, 79059456, 95292432, 51246720, 48847616, 14026752, 10321920

Offset

0,4

Comments

Row sums equal A007559(n+1), the triple factorial numbers.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1680 terms of this irregular triangle, as read by rows 0..40.

Formula

T(n,0) = n! for n >= 0.

T(n,2*n) = 2^n * n!, the even double factorials, for n >= 0.

Sum_{k=0..2*n} T(n,k) = Product_{k=0..n} (3*k + 1), the triple factorials, for n >= 0.

Example

This irregular triangle of coefficients T(n,k) of x^k in Product_{m=1..n} (m + x + 2*m*x^2), for n >= 0, k = 0..2*n, begins
1;
1, 1, 2;
2, 3, 9, 6, 8;
6, 11, 42, 45, 84, 44, 48;
24, 50, 227, 310, 717, 620, 908, 400, 384;
120, 274, 1425, 2277, 6165, 6917, 12330, 9108, 11400, 4384, 3840;
720, 1764, 10264, 18375, 56367, 74991, 154877, 149982, 225468, 147000, 164224, 56448, 46080;
5040, 13068, 83692, 163585, 556640, 838554, 1948268, 2254625, 3896536, 3354216, 4453120, 2617360, 2678144, 836352, 645120;
40320, 109584, 763244, 1601460, 5955777, 9882432, 25330938, 33402132, 64599201, 66804264, 101323752, 79059456, 95292432, 51246720, 48847616, 14026752, 10321920; ...
in which the central terms equal A322892:
[1, 1, 9, 45, 717, 6917, 154877, 2254625, 64599201, 1267075953, ...].
RELATED SEQUENCES.
Note that the terms in the secondary diagonal A322893 in this triangle,
[1, 3, 42, 310, 6165, 74991, 1948268, 33402132, 1070751825, ...],
may be divided by triangular numbers n*(n+1)/2 to obtain A322894:
[1, 1, 7, 31, 411, 3571, 69581, 927837, 23794485, 433057989, ...].

Prog

(PARI) {T(n, k) = polcoeff( prod(m=1, n, m + x + 2*m*x^2) +x*O(x^k), k)}
/* Print the irregular triangle */
for(n=0, 10, for(k=0, 2*n, print1( T(n, k), ", ")); print(""))

Crossrefs

Cf. A322892 (central terms), A322893 (diagonal), A322894.

Cf. A322235 (variant), A322225 (variant), A000165, A007559.

Sequence in context: A143307 A278463 A276248 * A275329 A021821 A019234

Adjacent sequences: A322888 A322889 A322890 * A322892 A322893 A322894

Keyword

nonn,tabf

Author

Paul D. Hanna, Dec 29 2018

Status

approved