A322225 - OEIS

%I #10 Dec 26 2018 10:40:34

%S 1,1,1,-1,2,3,-3,-3,2,6,11,-12,-21,12,11,-6,24,50,-61,-140,75,140,-61,

%T -50,24,120,274,-375,-1011,540,1475,-540,-1011,375,274,-120,720,1764,

%U -2696,-8085,4479,15456,-5005,-15456,4479,8085,-2696,-1764,720,5040,13068,-22148,-71639,42140,169266,-50932,-221389,50932,169266,-42140,-71639,22148,13068,-5040,40320,109584,-204436,-699804,442665,1969380,-575310,-3176172,593523,3176172,-575310,-1969380,442665,699804,-204436,-109584,40320,362880,1026576,-2093220,-7488928,5124105,24465321,-7192395,-46885278,7343325,57764619,-7343325,-46885278,7192395,24465321,-5124105,-7488928,2093220,1026576,-362880

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

%H Paul D. Hanna, <a href="/A322225/b322225.txt">Table of n, a(n) for n = 0..5040, as a flattened triangle of rows 0..70.</a>

%F Each row sums to 1.

%F Left and right borders equal n! and (-1)^n*n!, respectively.

%e This irregular triangle formed from coefficients of x^k in Product_{m=1..n} (m + x - m*x^2), for n >= 0, k = 0..2*n, begins

%e 1;

%e 1, 1, -1;

%e 2, 3, -3, -3, 2;

%e 6, 11, -12, -21, 12, 11, -6;

%e 24, 50, -61, -140, 75, 140, -61, -50, 24;

%e 120, 274, -375, -1011, 540, 1475, -540, -1011, 375, 274, -120;

%e 720, 1764, -2696, -8085, 4479, 15456, -5005, -15456, 4479, 8085, -2696, -1764, 720;

%e 5040, 13068, -22148, -71639, 42140, 169266, -50932, -221389, 50932, 169266, -42140, -71639, 22148, 13068, -5040;

%e 40320, 109584, -204436, -699804, 442665, 1969380, -575310, -3176172, 593523, 3176172, -575310, -1969380, 442665, 699804, -204436, -109584, 40320; ...

%e in which the central terms equal A322228.

%e RELATED SEQUENCES.

%e Note that the terms in the secondary diagonal A322227 in the above triangle

%e [1, 3, -12, -140, 540, 15456, -50932, -3176172, 7343325, 1053842295, ...]

%e may be divided by triangular numbers to obtain A322226:

%e [1, 1, -2, -14, 36, 736, -1819, -88227, 163185, 19160769, -15294993, ...].

%t row[n_] := CoefficientList[Product[k+x-k*x^2, {k, 1, n}] + O[x]^(2n+1), x];

%t Table[row[n], {n, 0, 9}] // Flatten (* _Jean-François Alcover_, Dec 26 2018 *)

%o (PARI) {T(n, k) = polcoeff( prod(m=1, n, m + x - m*x^2) +x*O(x^k), k)}

%o /* Print the irregular triangle */

%o for(n=0, 10, for(k=0, 2*n, print1( T(n, k), ", ")); print(""))

%Y Cf. A322226, A322227, A322228.

%Y Cf. A322235 (variant).

%K sign,tabf

%O 0,5

%A _Paul D. Hanna_, Dec 15 2018