A098437
Row sums in triangle of 3rd central factorial numbers (A098436).
Original entry on oeis.org
1, 2, 11, 111, 1732, 41153, 1361023, 59661972, 3400514643, 244686040585, 21672428066346, 2327934086035165, 299095824104595685, 45325168774732866658, 8011977427652269129031
Offset: 0
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{a(n)=polcoeff(sum(m=0, n, x^m/prod(k=1,m+1,1-k^3*x +x*O(x^n))), n)} /* Paul D. Hanna, Feb 15 2012 */
A269948
Triangle read by rows, Stirling set numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+k^3*T(n-1, k), for n>=0 and 0<=k<=n.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 1, 9, 1, 0, 1, 73, 36, 1, 0, 1, 585, 1045, 100, 1, 0, 1, 4681, 28800, 7445, 225, 1, 0, 1, 37449, 782281, 505280, 35570, 441, 1, 0, 1, 299593, 21159036, 33120201, 4951530, 130826, 784, 1
Offset: 0
1,
0, 1,
0, 1, 1,
0, 1, 9, 1,
0, 1, 73, 36, 1,
0, 1, 585, 1045, 100, 1,
0, 1, 4681, 28800, 7445, 225, 1,
0, 1, 37449, 782281, 505280, 35570, 441, 1.
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T := proc(n, k) option remember;
`if`(n=k, 1,
`if`(k<0 or k>n, 0,
T(n-1, k-1) + k^3*T(n-1, k))) end:
for n from 0 to 9 do seq(T(n,k), k=0..n) od;
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T[n_, n_] = 1; T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n - 1, k - 1] + k^3*T[n - 1, k]; T[, ] = 0;
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 25 2019 *)
A351800
a(n) = [x^n] 1/Product_{j=1..n} (1 - j^3*x).
Original entry on oeis.org
1, 1, 73, 28800, 33120201, 83648533275, 393764054984212, 3103381708489548640, 37965284782803741391413, 681476650259874114533077575, 17184647574689079046814198039765, 588057239856779143071625300022102376, 26548105106818292578525347802793561068860
Offset: 0
a(2) = (1*1)^3 + (1*2)^3 + (2*2)^3 = 1 + 8 + 64 = 73.
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b:= proc(n, k) option remember; `if`(k=0, 1,
add(b(j, k-1)*j^3, j=1..n))
end:
a:= n-> b(n$2):
seq(a(n), n=0..15);
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Table[SeriesCoefficient[Product[1/(1 - k^3*x), {k, 1, n}], {x, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, May 17 2025 *)
Showing 1-3 of 3 results.
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