A375597
Triangle read by rows: T(n, k) = n! * 3^k * hypergeom([-k], [-n], -2/3).
Original entry on oeis.org
1, 1, 1, 2, 4, 10, 6, 14, 34, 82, 24, 60, 152, 388, 1000, 120, 312, 816, 2144, 5656, 14968, 720, 1920, 5136, 13776, 37040, 99808, 269488, 5040, 13680, 37200, 101328, 276432, 755216, 2066032, 5659120, 40320, 110880, 305280, 841440, 2321664, 6412128, 17725952, 49045792, 135819136
Offset: 0
Triangle starts:
[0] 1;
[1] 1, 1;
[2] 2, 4, 10;
[3] 6, 14, 34, 82;
[4] 24, 60, 152, 388, 1000;
[5] 120, 312, 816, 2144, 5656, 14968;
[6] 720, 1920, 5136, 13776, 37040, 99808, 269488;
[7] 5040, 13680, 37200, 101328, 276432, 755216, 2066032, 5659120;
...
-
T[n_, k_] := (-2)^k*Sum[(-3/2)^(k - j)*Binomial[k, k - j]*(n - j)!, {j, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
A375612
Triangle read by rows: T(n, k) = n! * 4^k * hypergeom([-k], [-n], -1/4).
Original entry on oeis.org
1, 1, 3, 2, 7, 25, 6, 22, 81, 299, 24, 90, 338, 1271, 4785, 120, 456, 1734, 6598, 25121, 95699, 720, 2760, 10584, 40602, 155810, 598119, 2296777, 5040, 19440, 75000, 289416, 1117062, 4312438, 16651633, 64309755, 40320, 156240, 605520, 2347080, 9098904, 35278554, 136801778, 530555479, 2057912161
Offset: 0
Triangle starts:
[0] 1;
[1] 1, 3;
[2] 2, 7, 25;
[3] 6, 22, 81, 299;
[4] 24, 90, 338, 1271, 4785;
[5] 120, 456, 1734, 6598, 25121, 95699;
[6] 720, 2760, 10584, 40602, 155810, 598119, 2296777;
[7] 5040, 19440, 75000, 289416, 1117062, 4312438, 16651633, 64309755;
...
-
T[n_, k_] := (-1)^k*Sum[(-4)^(k - j)*Binomial[k, k - j]*(n - j)!, {j, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
A375613
Triangle read by rows: T(n, k) = n! * 4^k * hypergeom([-k], [-n], 1/4).
Original entry on oeis.org
1, 1, 5, 2, 9, 41, 6, 26, 113, 493, 24, 102, 434, 1849, 7889, 120, 504, 2118, 8906, 37473, 157781, 720, 3000, 12504, 52134, 217442, 907241, 3786745, 5040, 20880, 86520, 358584, 1486470, 6163322, 25560529, 106028861, 40320, 166320, 686160, 2831160, 11683224, 48219366, 199040786, 821723673, 3392923553
Offset: 0
Triangle starts:
[0] 1;
[1] 1, 5;
[2] 2, 9, 41;
[3] 6, 26, 113, 493;
[4] 24, 102, 434, 1849, 7889;
[5] 120, 504, 2118, 8906, 37473, 157781;
[6] 720, 3000, 12504, 52134, 217442, 907241, 3786745;
[7] 5040, 20880, 86520, 358584, 1486470, 6163322, 25560529, 106028861;
...
-
T[n_, k_] := Sum[4^(k - j)*Binomial[k, k - j]*(n - j)!, {j, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
Showing 1-3 of 3 results.