A383150
a(n) = Sum_{k=0..n} k^3 * (-1)^k * 3^(n-k) * binomial(n,k).
Original entry on oeis.org
0, -1, 2, 18, 64, 160, 288, 224, -1024, -6912, -28160, -95744, -294912, -851968, -2351104, -6266880, -16252928, -41222144, -102629376, -251527168, -608174080, -1453326336, -3437232128, -8055160832, -18723373056, -43201331200, -99019128832, -225586446336
Offset: 0
-
[2^(n-3) * (-12*n+9*n^2-n^3): n in [0..30]]; // Vincenzo Librandi, May 02 2025
-
Table[2^(n-3)*(-12*n+9*n^2-n^3),{n,0,30}] (* Vincenzo Librandi, May 02 2025 *)
-
a(n) = 2^(n-3)*(-12*n+9*n^2-n^3);
A383152
a(n) = Sum_{k=0..n} k^5 * (-1)^k * 3^(n-k) * binomial(n,k).
Original entry on oeis.org
0, -1, 26, 18, -272, -1400, -4032, -7168, -1024, 55296, 294400, 1086976, 3354624, 9132032, 22249472, 47923200, 85983232, 99155968, -102629376, -1237712896, -5688524800, -20775960576, -67868033024, -207022456832, -602167836672, -1690304512000, -4613767954432
Offset: 0
-
[&+[k^5 * (-1)^k * 3^(n-k) * Binomial(n, k): k in [0..n]]: n in [0..29]]; // Vincenzo Librandi, Apr 23 2025
-
Table[2^(n-5)*(-480*n+690*n^2-255*n^3+30*n^4-n^5),{n,0,50}] (* Vincenzo Librandi, Apr 24 2025 *)
-
a(n) = 2^(n-5)*(-480*n+690*n^2-255*n^3+30*n^4-n^5);
A383155
a(n) = Sum_{k=0..n} k^6 * (-1)^k * 3^(n-k) * binomial(n,k).
Original entry on oeis.org
0, -1, 58, -180, -1304, -2920, 1008, 34496, 163840, 525312, 1285120, 2241536, 1124352, -12113920, -72052736, -282378240, -924581888, -2699493376, -7201751040, -17666670592, -39507722240, -77918109696, -121883328512, -78622228480, 453588811776, 2904974950400, 11885785120768
Offset: 0
-
[&+[k^6 * (-1)^k * 3^(n-k) * Binomial(n,k): k in [0..n]]: n in [0..29]]; // Vincenzo Librandi, Apr 23 2025
-
Table[Sum[(k^6*(-1)^k*3^(n-k))*Binomial[n,k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Apr 23 2025 *)
-
a(n) = 2^(n-6)*(-4368*n+7290*n^2-3555*n^3+645*n^4-45*n^5+n^6);
A383149
Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) = (-1)^k * [m^k] (1/2^(m-n)) * Sum_{k=0..m} k^n * (-1)^m * 3^(m-k) * binomial(m,k).
Original entry on oeis.org
1, 0, 1, 0, 3, 1, 0, 12, 9, 1, 0, 66, 75, 18, 1, 0, 480, 690, 255, 30, 1, 0, 4368, 7290, 3555, 645, 45, 1, 0, 47712, 88536, 52290, 12705, 1365, 63, 1, 0, 608016, 1223628, 831684, 249585, 36120, 2562, 84, 1, 0, 8855040, 19019664, 14405580, 5073012, 915705, 87696, 4410, 108, 1
Offset: 0
f_0(m) = 1.
f_1(m) = -m.
f_2(m) = -3*m + m^2.
f_3(m) = -12*m + 9*m^2 - m^3.
f_4(m) = -66*m + 75*m^2 - 18*m^3 + m^4.
f_5(m) = -480*m + 690*m^2 - 255*m^3 + 30*m^4 - m^5.
Triangle begins:
1;
0, 1;
0, 3, 1;
0, 12, 9, 1;
0, 66, 75, 18, 1;
0, 480, 690, 255, 30, 1;
0, 4368, 7290, 3555, 645, 45, 1;
0, 47712, 88536, 52290, 12705, 1365, 63, 1;
...
-
T(n, k) = sum(j=k, n, 2^(n-j)*stirling(n, j, 2)*abs(stirling(j, k, 1)));
-
def a_row(n):
s = sum(2^(n-k)*stirling_number2(n, k)*rising_factorial(x, k) for k in (0..n))
return expand(s).list()
for n in (0..9): print(a_row(n))
Showing 1-4 of 4 results.