A269947
Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+(n-1)^3*T(n-1,k), for n>=0 and 0<=k<=n.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 8, 9, 1, 0, 216, 251, 36, 1, 0, 13824, 16280, 2555, 100, 1, 0, 1728000, 2048824, 335655, 15055, 225, 1, 0, 373248000, 444273984, 74550304, 3587535, 63655, 441, 1, 0, 128024064000, 152759224512, 26015028256, 1305074809, 25421200, 214918, 784, 1
Offset: 0
Triangle starts:
1,
0, 1,
0, 1, 1,
0, 8, 9, 1,
0, 216, 251, 36, 1,
0, 13824, 16280, 2555, 100, 1,
0, 1728000, 2048824, 335655, 15055, 225, 1.
-
T := proc(n, k) option remember;
`if`(n=k, 1,
`if`(k<0 or k>n, 0,
T(n-1, k-1) + (n-1)^3*T(n-1, k))) end:
for n from 0 to 6 do seq(T(n,k), k=0..n) od;
-
T[n_, k_] := T[n, k] = Which[n == k, 1, k < 0 || k > n, 0, True, T[n - 1, k - 1] + (n - 1)^3 T[n - 1, k]];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 12 2019 *)
A107415
Triangle, read by rows: T(0,0) = 1; T(n,k) = n!*T(n-1,k) - T(n-1,k-1).
Original entry on oeis.org
1, 1, -1, 2, -3, 1, 12, -20, 9, -1, 288, -492, 236, -33, 1, 34560, -59328, 28812, -4196, 153, -1, 24883200, -42750720, 20803968, -3049932, 114356, -873, 1, 125411328000, -215488512000, 104894749440, -15392461248, 579404172, -4514276, 5913, -1
Offset: 0
Triangle begins
1;
1, -1;
2, -3, 1;
12, -20, 9, -1;
288, -492, 236, -33, 1;
34560, -59328, 28812, -4196, 153, -1;
24883200, -42750720, 20803968, -3049932, 114356, -873, 1;
(1 - x) * (2 - x) = 2 - 3*x + x^2, (1 - x) * (2 - x) * (6 - x) = 12 - 20*x + 9*x^2 - x^3, etc. - _Seiichi Manyama_, Sep 24 2021
-
t(n, k) = {if (k < 0, return (0)); if (n < k, return (0)); if (n == 0, return (1)); return (n!*t(n-1, k) - t(n-1, k-1));} \\ Michel Marcus, Apr 11 2013
-
row(n) = Vecrev(prod(k=1, n, k!-x)); \\ Seiichi Manyama, Sep 24 2021
A348014
Triangle, read by rows, with row n forming the coefficients in Product_{k=0..n} (1 + k^k*x).
Original entry on oeis.org
1, 1, 1, 1, 5, 4, 1, 32, 139, 108, 1, 288, 8331, 35692, 27648, 1, 3413, 908331, 26070067, 111565148, 86400000, 1, 50069, 160145259, 42405161203, 1216436611100, 5205269945088, 4031078400000
Offset: 0
Triangle begins:
1;
1, 1;
1, 5, 4;
1, 32, 139, 108;
1, 288, 8331, 35692, 27648;
1, 3413, 908331, 26070067, 111565148, 86400000;
The diagonal of the triangle is
A002109.
-
T(n, k) = if(k==0, 1, if(k==n, prod(j=1, n, j^j), T(n-1, k)+n^n*T(n-1, k-1)));
-
row(n) = Vecrev(prod(k=1, n, 1+k^k*x));
Showing 1-3 of 3 results.
Comments