A374428
Triangle read by rows: T(n, k) = n! * 2^k * hypergeom([-k], [-n], 1/2).
Original entry on oeis.org
1, 1, 3, 2, 5, 13, 6, 14, 33, 79, 24, 54, 122, 277, 633, 120, 264, 582, 1286, 2849, 6331, 720, 1560, 3384, 7350, 15986, 34821, 75973, 5040, 10800, 23160, 49704, 106758, 229502, 493825, 1063623, 40320, 85680, 182160, 387480, 824664, 1756086, 3741674, 7977173, 17017969
Offset: 0
-
T[n_, k_] := n! 2^k Hypergeometric1F1[-k, -n, 1/2];
(* Alternative: *)
T[n_, k_] := Sum[2^(k - j)*Binomial[k, k - j]*((n - j)!), {j, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Detlef Meya, Aug 12 2024 *)
A375446
Triangle read by rows: T(n, k) = n! * 3^k * hypergeom([-k], [-n], -1/3).
Original entry on oeis.org
1, 1, 2, 2, 5, 13, 6, 16, 43, 116, 24, 66, 182, 503, 1393, 120, 336, 942, 2644, 7429, 20894, 720, 2040, 5784, 16410, 46586, 132329, 376093, 5040, 14400, 41160, 117696, 336678, 963448, 2758015, 7897952, 40320, 115920, 333360, 958920, 2759064, 7940514, 22858094, 65816267, 189550849
Offset: 0
Triangle starts:
[0] 1,
[1] 1, 2,
[2] 2, 5, 13,
[3] 6, 16, 43, 116,
[4] 24, 66, 182, 503, 1393,
[5] 120, 336, 942, 2644, 7429, 20894,
[6] 720, 2040, 5784, 16410, 46586, 132329, 376093,
[7] 5040, 14400, 41160, 117696, 336678, 963448, 2758015, 7897952,
...
-
T[n_, k_] := (-1)^k*Sum[(-3)^(k - j)*Binomial[k, k - j]*(n - j)!, {j, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
A375447
Triangle read by rows: T(n, k) = n! * 3^k * hypergeom([-k], [-n], 1/3).
Original entry on oeis.org
1, 1, 4, 2, 7, 25, 6, 20, 67, 226, 24, 78, 254, 829, 2713, 120, 384, 1230, 3944, 12661, 40696, 720, 2280, 7224, 22902, 72650, 230611, 732529, 5040, 15840, 49800, 156624, 492774, 1550972, 4883527, 15383110, 40320, 126000, 393840, 1231320, 3850584, 12044526, 37684550, 117937177, 369194641
Offset: 0
Triangle starts:
[0] 1;
[1] 1, 4;
[2] 2, 7, 25;
[3] 6, 20, 67, 226;
[4] 24, 78, 254, 829, 2713;
[5] 120, 384, 1230, 3944, 12661, 40696;
[6] 720, 2280, 7224, 22902, 72650, 230611, 732529;
-
T[n_, k_] := Sum[3^(k - j)*Binomial[k, k - j]*((n - j)!), {j, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
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
A375600
Triangle read by rows: T(n, k) = n! * 3^k * hypergeom([-k], [-n], 2/3).
Original entry on oeis.org
1, 1, 5, 2, 8, 34, 6, 22, 82, 314, 24, 84, 296, 1052, 3784, 120, 408, 1392, 4768, 16408, 56792, 720, 2400, 8016, 26832, 90032, 302912, 1022320, 5040, 16560, 54480, 179472, 592080, 1956304, 6474736, 21468848, 40320, 131040, 426240, 1387680, 4521984, 14750112, 48162944, 157438304, 515252608
Offset: 0
Triangle starts:
[0] 1;
[1] 1, 5;
[2] 2, 8, 34;
[3] 6, 22, 82, 314;
[4] 24, 84, 296, 1052, 3784;
[5] 120, 408, 1392, 4768, 16408, 56792;
[6] 720, 2400, 8016, 26832, 90032, 302912, 1022320;
[7] 5040, 16560, 54480, 179472, 592080, 1956304, 6474736, 21468848;
...
-
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 1-7 of 7 results.