A132166 A convolution triangle of numbers obtained from A036224.
1, 21, 1, 336, 42, 1, 4536, 1113, 63, 1, 54432, 23184, 2331, 84, 1, 598752, 412272, 65205, 3990, 105, 1, 6158592, 6531840, 1518048, 139860, 6090, 126, 1, 60046272, 94618368, 30912840, 4010769, 256410, 8631, 147, 1, 560431872, 1274921856
Offset: 1
Examples
{1};{21,1};{336,42,1};{4536,1113,63,1};...; Row polynomial s(3,x)=336*x+42*x^2+x^3.
Links
- W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
- W. Lang, First ten rows.
Crossrefs
Formula
a(n, m) = 6*(6*m+n-1)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n
G.f. for m-th column: ((1-(1-6*x)^6)/(36*(1-6*x)^6))^m.
A053110 Expansion of (-1 + 1/(1-7*x)^7)/(49*x); related to A036226.
1, 28, 588, 10290, 158466, 2218524, 28840812, 353299947, 4121832715, 46164526408, 499416240232, 5243870522436, 53648829191076, 536488291910760, 5257585260725448, 50604258134482437, 479252091744216021
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..400
- Index entries for linear recurrences with constant coefficients, signature (49,-1029,12005,-84035,352947,-823543,823543).
Programs
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Magma
[7^(n-1)*Binomial(n+7, 6): n in [0..30]]; // G. C. Greubel, Aug 16 2018
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Mathematica
CoefficientList[Series[(-1+1/(1-7x)^7)/(49x),{x,0,30}],x] (* or *) LinearRecurrence[{49,-1029,12005,-84035,352947,-823543,823543},{1,28,588,10290,158466,2218524,28840812},30] (* Harvey P. Dale, Jun 03 2015 *) Table[7^(n-1)*Binomial[n+7, 6], {n,0,30}] (* G. C. Greubel, Aug 16 2018 *) -
PARI
vector(30,n,n--; 7^(n-1)*binomial(n+7, 6)) \\ G. C. Greubel, Aug 16 2018
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Sage
[lucas_number2(n, 7, 0)*binomial(n,6)/7^8 for n in range(7, 24)] # Zerinvary Lajos, Mar 13 2009
Formula
a(n) = 7^(n-1)*binomial(n+7, 6);
G.f.: (-1 + (1-7*x)^(-7))/(x*7^2).
A053113 Expansion of (-1 + 1/(1-10*x)^10)/(100*x); related to A053109.
1, 55, 2200, 71500, 2002000, 50050000, 1144000000, 24310000000, 486200000000, 9237800000000, 167960000000000, 2939300000000000, 49742000000000000, 817190000000000000, 13075040000000000000, 204297500000000000000
Offset: 0
Comments
Links
- G. C. Greubel, Table of n, a(n) for n = 0..400
- W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
- Index entries for linear recurrences with constant coefficients, signature (100, -4500, 120000, -2100000, 25200000, -210000000, 1200000000, -4500000000, 10000000000, -10000000000).
Programs
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Magma
[10^(n-1)*Binomial(n+10, 9): n in [0..30]]; // G. C. Greubel, Aug 16 2018
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Mathematica
Table[10^(n - 1)*Binomial[n + 10, 9], {n, 0, 30}] (* G. C. Greubel, Aug 16 2018 *) LinearRecurrence[{100,-4500,120000,-2100000,25200000,-210000000,1200000000,-4500000000,10000000000,-10000000000},{1,55,2200,71500,2002000,50050000,1144000000,24310000000,486200000000,9237800000000},20] (* Harvey P. Dale, Jul 30 2025 *) -
PARI
vector(30,n,n--; 10^(n-1)*binomial(n+10, 9)) \\ G. C. Greubel, Aug 16 2018
Formula
a(n) = 10^(n-1)*binomial(n+10, 9).
G.f.: (-1 + (1-10*x)^(-10))/(x*10^2).
Comments