A091466
Table (by antidiagonals) of labeled alternating octopuses with n black nodes and k white nodes. Each type of object labeled from its own label set.
Original entry on oeis.org
1, 2, 2, 0, 14, 0, 0, 36, 36, 0, 0, 24, 336, 24, 0, 0, 0, 1296, 1296, 0, 0, 0, 0, 2160, 14832, 2160, 0, 0, 0, 0, 1440, 80640, 80640, 1440, 0, 0, 0, 0, 0, 233280, 1082880, 233280, 0, 0, 0, 0, 0, 0, 362880, 7776000, 7776000, 362880, 0, 0, 0, 0, 0, 0, 241920, 33264000
Offset: 1
1 2 0 0 0 ...
2 4 36 24 0 ...
0 36 336 1296 2160 ...
0 24 1296 14832 80640 ...
0 0 2160 80640 1082880 ...
- F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 113 (2.4.41).
A156815
Triangle T(n, k) = n!*StirlingS2(n, k)/binomial(n, k), read by rows.
Original entry on oeis.org
1, 0, 1, 0, 1, 2, 0, 2, 6, 6, 0, 6, 28, 36, 24, 0, 24, 180, 300, 240, 120, 0, 120, 1488, 3240, 3120, 1800, 720, 0, 720, 15120, 43344, 50400, 33600, 15120, 5040, 0, 5040, 182880, 695520, 979776, 756000, 383040, 141120, 40320, 0, 40320, 2570400, 13068000, 22377600, 20018880, 11430720, 4656960, 1451520, 362880
Offset: 0
Triangle begins as:
1;
0, 1;
0, 1, 2;
0, 2, 6, 6;
0, 6, 28, 36, 24;
0, 24, 180, 300, 240, 120;
0, 120, 1488, 3240, 3120, 1800, 720;
0, 720, 15120, 43344, 50400, 33600, 15120, 5040;
0, 5040, 182880, 695520, 979776, 756000, 383040, 141120, 40320;
- Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), page 99.
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[Factorial(n)*StirlingSecond(n,k)/Binomial(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 10 2021
-
T[n_, k_] = n!*StirlingS2[n, k]/Binomial[n, k];
Table[T[n, k], {n, 0, 12}, {k,0,n}]//Flatten
-
flatten([[factorial(n)*stirling_number2(n,k)/binomial(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 10 2021
A371685
Triangle read by rows: T(n, k) = n! * Sum_{j=0..n-1} binomial(k - 1, j) / (j + 1).
Original entry on oeis.org
0, 1, 1, 1, 2, 3, 5, 6, 9, 14, 14, 24, 36, 56, 90, 94, 120, 180, 280, 450, 744, 444, 720, 1080, 1680, 2700, 4464, 7560, 3828, 5040, 7560, 11760, 18900, 31248, 52920, 91440, 25584, 40320, 60480, 94080, 151200, 249984, 423360, 731520, 1285200
Offset: 0
Triangle starts:
[0] 0;
[1] 1, 1;
[2] 1, 2, 3;
[3] 5, 6, 9, 14;
[4] 14, 24, 36, 56, 90;
[5] 94, 120, 180, 280, 450, 744;
[6] 444, 720, 1080, 1680, 2700, 4464, 7560;
[7] 3828, 5040, 7560, 11760, 18900, 31248, 52920, 91440;
-
T := (n, k) -> local j; n!*add(binomial(k-1, j)/(j + 1), j = 0..n-1):
T := (n, k) -> local j; n!*ifelse(n = 0, 0, ifelse(k=0, add(-(-1)^j/j, j = 1..n), (2^k - 1) / k)):
seq(print(seq(T(n, k), k = 0..n)), n = 0..7);
A371762
Triangle read by rows: the polynomial coefficients of the numerator of the rational solution of the linear recurrence equations of the rows of A371761.
Original entry on oeis.org
1, 0, 1, 0, 1, 2, 0, 1, 7, 6, 0, 1, 19, 46, 24, 0, 1, 46, 251, 326, 120, 0, 1, 104, 1163, 3016, 2556, 720, 0, 1, 225, 4831, 23283, 35848, 22212, 5040, 0, 1, 473, 18523, 158531, 417148, 437228, 212976, 40320, 0, 1, 976, 66886, 976636, 4285549, 7084804, 5586444, 2239344, 362880
Offset: 0
Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 1, 2;
[3] 0, 1, 7, 6;
[4] 0, 1, 19, 46, 24;
[5] 0, 1, 46, 251, 326, 120;
[6] 0, 1, 104, 1163, 3016, 2556, 720;
[7] 0, 1, 225, 4831, 23283, 35848, 22212, 5040;
[8] 0, 1, 473, 18523, 158531, 417148, 437228, 212976, 40320;
[9] 0, 1, 976, 66886, 976636, 4285549, 7084804, 5586444, 2239344, 362880;
.
The rational generating function for row 3 of A371761 is:
gf = (6*x^3 + 7*x^2 + x)/(-6*x^3 + 11*x^2 - 6*x + 1).
A366953
Expansion of e.g.f. -log(1 - x^2/(1-x)^3).
Original entry on oeis.org
0, 0, 2, 18, 156, 1560, 18600, 262080, 4284000, 79833600, 1671788160, 38878963200, 994367404800, 27741377664000, 838406080512000, 27287153858764800, 951529253875200000, 35392677532692480000, 1398727294865252352000, 58529661706809925632000, 2585234639710708457472000
Offset: 0
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