A374427
Triangle read by rows: T(n, k) = n! * 2^k * hypergeom([-k], [-n], -1/2).
Original entry on oeis.org
1, 1, 1, 2, 3, 5, 6, 10, 17, 29, 24, 42, 74, 131, 233, 120, 216, 390, 706, 1281, 2329, 720, 1320, 2424, 4458, 8210, 15139, 27949, 5040, 9360, 17400, 32376, 60294, 112378, 209617, 391285, 40320, 75600, 141840, 266280, 500184, 940074, 1767770, 3325923, 6260561
Offset: 0
1
1 1
2 3 5
6 10 17 29
24 42 74 131 233
120 216 390 706 1281 2329
720 1320 2424 4458 8210 15139 27949
5040 9360 17400 32376 60294 112378 209617 391285
40320 75600 141840 266280 500184 940074 1767770 3325923 6260561
362880 685440 1295280 2448720 4631160 8762136 16584198 31400626 59475329
-
A374427 := proc(n,k)
(-1)^k*add((-2)^(k-j)*binomial(k,k-j)*(n-j)!,j=0..k) ;
end proc:
seq(seq(A374427(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Aug 30 2024
-
T[n_, k_] := n! 2^k Hypergeometric1F1[-k, -n, -1/2];
(* Alternative: )
T[n_, k_] := (-1)^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.