A049224 A convolution triangle of numbers obtained from A025751.
1, 15, 1, 330, 30, 1, 8415, 885, 45, 1, 232254, 26730, 1665, 60, 1, 6735366, 825858, 58320, 2670, 75, 1, 202060980, 25992252, 2003562, 106560, 3900, 90, 1, 6213375135, 830282805, 68351283, 4038741, 174825, 5355, 105, 1, 194685754230
Offset: 1
Links
- W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Programs
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Maxima
T(n,m):=(m*sum(binomial(-m+2*i-1,i-1)*2^(2*n-2*i)*sum(binomial(k,n-k-i)*3^(k+i-m)*(-1)^(n-k-i)*binomial(n+k-1,n-1),k,0,n-i),i,m,n))/n; /* Vladimir Kruchinin, Dec 21 2011 */
Formula
a(n, m) = 6*(6*(n-1)-m)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n
G.f.: [(1-(1-36*x)^(1/6))/6]^m=sum(n>=m, T(n,m)*x^n), T(n,m)=(m*sum(i=m..n, binomial(-m+2*i-1,i-1)*2^(2*n-2*i)*sum(k=0..n-i, binomial(k,n-k-i)*3^(k+i-m)*(-1)^(n-k-i)*binomial(n+k-1,n-1))))/n. - Vladimir Kruchinin, Dec 21 2011
A248328 Square array read by antidiagonals downwards: super Patalan numbers of order 6.
1, 6, 30, 126, 90, 990, 3276, 1260, 1980, 33660, 93366, 24570, 20790, 50490, 1161270, 2800980, 560196, 324324, 424116, 1393524, 40412196, 86830380, 14004900, 6162156, 5513508, 9754668, 40412196, 1414426860, 2753763480, 372130200, 132046200, 89791416, 108694872, 242473176, 1212365880
Offset: 0
Comments
Examples
T(0..4,0..4) is 1 6 126 3276 93366 30 90 1260 24570 560196 990 1980 20790 324324 6162156 33660 50490 424116 5513508 89791416 1161270 1393524 9754668 108694872 1548901926
Links
- Thomas M. Richardson, The Super Patalan Numbers, arXiv:1410.5880 [math.CO], 2014.
- Thomas M. Richardson, The Super Patalan Numbers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.3.
Crossrefs
Programs
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PARI
matrix(5, 5, nn, kk, n=nn-1;k=kk-1;(-1)^k*36^(n+k)*binomial(n-1/6,n+k)) \\ Michel Marcus, Oct 09 2014
Formula
T(0,0)=1, T(n,k) = T(n-1,k)*(36*n-6)/(n+k), T(n,k) = T(n,k-1)*(36*k-30)/(n+k).
G.f.: (x/(1-36*x)^(5/6)+y/(1-36*y)^(1/6))/(x+y-36*x*y).
T(n,k) = (-1)^k*36^(n+k)*binomial(n-1/6,n+k).
A034789 Related to sextic factorial numbers A008542.
1, 21, 546, 15561, 466830, 14471730, 458960580, 14801478705, 483514971030, 15955994043990, 530899438190940, 17785131179396490, 599222112044281740, 20287948650642110340, 689790254121831751560, 23539092421907508521985, 805867752326480585870310, 27668126163209166781547310
Offset: 1
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..645
- Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
- Index entries for sequences related to factorial numbers.
Programs
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GAP
List([1..20], n-> 6^(n-1)*Product([1..n], j-> 6*j-5)/Factorial(n) ); # G. C. Greubel, Nov 11 2019
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Magma
[6^(n-1)*(&*[6*j-5: j in [1..n]])/Factorial(n): n in [1..20]]; // G. C. Greubel, Nov 11 2019
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Maple
seq( 6^(n-1)*mul(6*j-5, j=1..n)/n!, n=1..20); # G. C. Greubel, Nov 11 2019
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Mathematica
Rest@ CoefficientList[Series[(-1 + (1 - 36 x)^(-1/6))/6, {x, 0, 16}], x] (* Michael De Vlieger, Oct 13 2019 *) Table[6^(2*n-1)*Pochhammer[1/6, n]/n!, {n, 20}] (* G. C. Greubel, Nov 11 2019 *) -
PARI
vector(20, n, 6^(n-1)*prod(j=1,n, 6*j-5)/n! ) \\ G. C. Greubel, Nov 11 2019
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Sage
[6^(n-1)*product( (6*j-5) for j in (1..n))/factorial(n) for n in (1..20)] # G. C. Greubel, Nov 11 2019
Formula
a(n) = 6^(n-1)*A008542(n)/n!.
G.f.: (-1+(1-36*x)^(-1/6))/6.
D-finite with recurrence: n*a(n) + 6*(-6*n+5)*a(n-1) = 0. - R. J. Mathar, Jan 28 2020
a(n) ~ 6^(2*n-1) * n^(-5/6) / Gamma(1/6). - Amiram Eldar, Aug 18 2025
Comments