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A322892 a(n) = [x^n] Product_{k=1..n} (k + x + 2*k*x^2), for n >= 0. 4
1, 1, 9, 45, 717, 6917, 154877, 2254625, 64599201, 1267075953, 44097148953, 1092097482333, 44645622936189, 1338624157833861, 62791851488870493, 2213430779241737793, 117082536584478235713, 4748345510312622896993, 279463602946698380026793, 12824987274099379222626701, 830920299335152521399853101, 42586722790649923167650932101, 3011022417317079016258969826109, 170527854080899363788154404878305 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..300

FORMULA

a(n+1) = 4*(n+1) * A322893(n) + a(n), for n >= 1.

a(n+1) = 2*n*(n+1)^2 * A322894(n) + a(n), for n >= 1.

EXAMPLE

The irregular triangle A322891 of coefficients 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; ...

in which the main diagonal forms this sequence.

Note that the terms in the secondary diagonal A322893 in the above 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) {A322891(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( A322891(n, k), ", ")); print(""))

/* Print this sequence */

for(n=0, 30, print1( A322891(n, n), ", "))

CROSSREFS

Cf. A322891, A322893, A322894.

Cf. A322238 (variant).

Sequence in context: A058824 A244302 A177742 * A355175 A352398 A261847

Adjacent sequences:  A322889 A322890 A322891 * A322893 A322894 A322895

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 29 2018

STATUS

approved

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Last modified October 4 21:22 EDT 2022. Contains 357240 sequences. (Running on oeis4.)