A325873
T(n, k) = [x^k] Sum_{k=0..n} |Stirling1(n, k)|*FallingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.
Original entry on oeis.org
1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 4, 0, 1, 0, 8, 5, 10, 0, 1, 0, 26, 58, 15, 20, 0, 1, 0, 194, 217, 238, 35, 35, 0, 1, 0, 1142, 2035, 1008, 728, 70, 56, 0, 1, 0, 9736, 13470, 11611, 3444, 1848, 126, 84, 0, 1, 0, 81384, 134164, 85410, 47815, 9660, 4116, 210, 120, 0, 1
Offset: 0
Triangle starts:
[0] [1]
[1] [0, 1]
[2] [0, 0, 1]
[3] [0, 1, 0, 1]
[4] [0, 1, 4, 0, 1]
[5] [0, 8, 5, 10, 0, 1]
[6] [0, 26, 58, 15, 20, 0, 1]
[7] [0, 194, 217, 238, 35, 35, 0, 1]
[8] [0, 1142, 2035, 1008, 728, 70, 56, 0, 1]
[9] [0, 9736, 13470, 11611, 3444, 1848, 126, 84, 0, 1]
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p[n_] := Sum[Abs[StirlingS1[n, k]] FactorialPower[x, k], {k, 0, n}];
Table[CoefficientList[FunctionExpand[p[n]], x], {n, 0, 9}] // Flatten
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T(n, k) = sum(j=k, n, abs(stirling(n, j, 1))*stirling(j, k, 1)); \\ Seiichi Manyama, Apr 18 2025
-
def a_row(n):
s = sum(stirling_number1(n,k)*falling_factorial(x,k) for k in (0..n))
return expand(s).list()
[a_row(n) for n in (0..10)]
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))
A326717
Coefficients of polynomials related to ordered set partitions. Triangle read by rows, T_{m}(n, k) for m = 5 and 0 <= k <= n.
Original entry on oeis.org
1, 0, 1, 0, 127, 126, 0, 255256, 381381, 126126, 0, 2979852651, 5447453786, 2956465512, 488864376, 0, 127156445503275, 264284637872750, 184292523727620, 52359004217520, 5194672859376
Offset: 0
Triangle starts:
[0] [1]
[1] [0, 1]
[2] [0, 127, 126]
[3] [0, 255256, 381381, 126126]
[4] [0, 2979852651, 5447453786, 2956465512, 488864376]
[5] [0, 127156445503275, 264284637872750, 184292523727620, 52359004217520, 5194672859376]
[6] [0, 15160169962750251082, 34544220081315967665, 28276496764200664980, 10634436034307385300, 1865368063755476280, 123378675083039376]