A036068 -id:A036068 - cp's OEIS frontend

A030524 A convolution triangle of numbers obtained from A036068.

1, 6, 1, 30, 12, 1, 135, 96, 18, 1, 567, 630, 198, 24, 1, 2268, 3654, 1701, 336, 30, 1, 8748, 19440, 12501, 3564, 510, 36, 1, 32805, 96957, 82296, 31644, 6435, 720, 42, 1, 120285, 459756, 498663, 247536, 66915, 10530, 966, 48, 1, 433026, 2092959

Offset: 1

Wolfdieter Lang

a(n,m) := s1p(4; n,m), a member of a sequence of unsigned triangles including s1p(2; n,m)= A007318(n-1,m-1) (Pascal's triangle). Signed version: (-1)^(n-m)*a(n,m) := s1(4; n,m).

			{1}; {6,1}; {30,12,1}; {135,96,18,1}; {567,630,198,24,1}; ...

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Cf. A030523, A043553. a(n, 1)= A036068(n-1). Row sums = A043553(n).

a(n, m) = 3*(3*m+n-1)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n

A033842 Triangle of coefficients of certain polynomials (exponents in decreasing order).

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 16, 16, 6, 1, 125, 125, 50, 10, 1, 1296, 1296, 540, 120, 15, 1, 16807, 16807, 7203, 1715, 245, 21, 1, 262144, 262144, 114688, 28672, 4480, 448, 28, 1, 4782969, 4782969, 2125764, 551124, 91854, 10206, 756, 36, 1, 100000000

Offset: 0

Wolfdieter Lang

See A049323.

			{1}; {1,1}; {3,3,1}; {16,16,6,1}; {125,125,50,10,1}; .... E.g. third row {3,3,1} corresponds to polynomial p(2,x)= 3*x^2+3*x+1.

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Thierry Lévy, The Number of Prefixes of Minimal Factorisations of a Cycle, The Electronic Journal of Combinatorics, 23(3) (2016), #P3.35

a(n, 0)= A000272(n+1), n >= 0 (first column), a(n, 1)= A000272(n+1), n >= 1 (second column). p(k-1, -x)/(1-k*x)^k = (-1+1/(1-k*x)^k)/(x*k^2) is for k=1..5 G.f. for A000012, A001792, A036068, A036070, A036083, respectively.

A080421 a(n) = (n+1)(n+2)(n+9)*3^n/18.

Original entry on oeis.org

1, 10, 66, 360, 1755, 7938, 34020, 139968, 557685, 2165130, 8227494, 30705480, 112842639, 409209570, 1466777160, 5203870272, 18294856425, 63795240522, 220829678730, 759344158440, 2595329855811, 8821564534530, 29832927334956, 100419390748800, 336561864306525

Offset: 0

Paul Barry, Feb 19 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-54,108,-81).

T(n,3) in triangle A080419.

Cf. A036068, A136158.

[(n+1)*(n+2)*(n+9)*3^n/18: n in [0..30]]; // Vincenzo Librandi, Aug 05 2013

Table[((n+1)(n+2)(n+9)3^n)/18,{n,0,30}] (* or *) LinearRecurrence[ {12,-54,108,-81},{1,10,66,360},30] (* Harvey P. Dale, Mar 21 2012 *)
CoefficientList[Series[(1 - 2 x) / (1 - 3 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 05 2013 *)

[(n+1)*(n+2)*(n+9)*3^(n-2)/2 for n in range(31)] # G. C. Greubel, Dec 22 2023

G.f.: (1-2*x)/(1-3*x)^4.
a(n) = A006503(n+1)*3^(n-1).
a(n) = 12*a(n-1)-54*a(n-2)+108*a(n-3)-81*a(n-4). - Harvey P. Dale, Mar 21 2012
From G. C. Greubel, Dec 22 2023: (Start)
a(n) = (n+9)*A036068(n-1).
a(n) = A136158(n+3, 3).
E.g.f.: (1/2)*(2 + 14*x + 15*x^2 + 3*x^3)*exp(3*x). (End)
From Amiram Eldar, Jan 11 2024: (Start)
Sum_{n>=0} 1/a(n) = 44172*log(3/2)/7 - 20050659/7840.
Sum_{n>=0} (-1)^n/a(n) = 44496*log(4/3)/7 - 14329629/7840. (End)

A053113 Expansion of (-1 + 1/(1-10x)^10)/(100x); related to A053109.

Original entry on oeis.org

1, 55, 2200, 71500, 2002000, 50050000, 1144000000, 24310000000, 486200000000, 9237800000000, 167960000000000, 2939300000000000, 49742000000000000, 817190000000000000, 13075040000000000000, 204297500000000000000

Offset: 0

Wolfdieter Lang

This is the tenth member of the k-family of sequences a(k,n) := k^(n-1)*binomial(n+k,k-1) starting with A000012 (powers of 1), A001792, A036068, A036070, A036083, A036224, A053110-113 for k=1..10.

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).

[10^(n-1)*Binomial(n+10, 9): n in [0..30]]; // G. C. Greubel, Aug 16 2018

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 *)

vector(30,n,n--; 10^(n-1)*binomial(n+10, 9)) \\ G. C. Greubel, Aug 16 2018

a(n) = 10^(n-1)*binomial(n+10, 9).

G.f.: (-1 + (1-10*x)^(-10))/(x*10^2).

A132166 A convolution triangle of numbers obtained from A036224.

Original entry on oeis.org

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

Wolfdieter Lang, Oct 12 2007

Signed version: (-1)^(n-m)*a(n, m) := s1(7; n,m).

a(n,m) := s1p(7; n,m), a member of a sequence of unsigned triangles including s1p(2; n,m)= A007318(n-1,m-1) (Pascal's triangle), A030523=s1p(3), A036068=s1p(4), A030526=s1p(5) and A030527=s1p(6).

			{1};{21,1};{336,42,1};{4536,1113,63,1};...; Row polynomial s(3,x)=336*x+42*x^2+x^3.

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
W. Lang, First ten rows.

Related triangle A134141 (S1p(7)).

Cf. A036224(n-1), n>=1 (first column). A132167 (row sums). A132168 (alternating row sums).

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.

A049323 Triangle of coefficients of certain polynomials (exponents in increasing order), equivalent to A033842.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 6, 16, 16, 1, 10, 50, 125, 125, 1, 15, 120, 540, 1296, 1296, 1, 21, 245, 1715, 7203, 16807, 16807, 1, 28, 448, 4480, 28672, 114688, 262144, 262144, 1, 36, 756, 10206, 91854, 551124, 2125764, 4782969, 4782969, 1, 45, 1200, 21000, 252000

Offset: 0

Wolfdieter Lang

These polynomials p(n, x) appear in the W. Lang reference as c1(-(n+1);x), n >= 0 on p.12. The coefficients are given there in eq.(44) on p. 6. - Wolfdieter Lang, Nov 20 2015

			The triangle a(n, m) begins:
n\m 0  1   2    3     4      5      6      7 ...
0:  1
1:  1  1
2:  1  3   3
3:  1  6  16   16
4:  1 10  50  125  125
5:  1 15 120  540  1296  1296
6:  1 21 245 1715  7203  16807  16807
7:  1 28 448 4480 28672 114688 262144 262144
... reformatted. - Wolfdieter Lang, Nov 20 2015
E.g. the third row {1,3,3} corresponds to polynomial p(2,x)= 1 + 3*x + 3*x^2.

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

a(n, 0)= A000012 (powers of 1), a(n, 1)= A000217 (triangular numbers), a(n, n)= A000272(n+1), n >= 0 (diagonal), a(n, n-1)= A000272(n+1), n >= 1.
For n = 0..5 the row sequences a(n, m), m >= 0, are the first columns of the triangles A023531 (unit matrix), A030528, A049324, A049325, A049326, A049327, respectively.
Cf. A033842, A046757.

/* As triangle: */ [[Binomial(n+1, k+1)*(n+1)^(k-1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Nov 20 2015

seq(seq(binomial(n+1,m+1)*(n+1)^(m-1),m=0..n),n=0..10); # Robert Israel, Oct 19 2015

Table[Binomial[n + 1, k + 1] (n + 1)^(k - 1), {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Nov 19 2015 *)

a(n, m) = A033842(n, n-m) = binomial(n+1, m+1)*(n+1)^{m-1}, n >= m >= 0, else 0.
p(k-1, -x)/(1-k*x)^k =(-1+1/(1-k*x)^k)/(x*k^2) is for k=1..5 G.f. for A000012, A001792, A036068, A036070, A036083, respectively.
From Werner Schulte, Oct 19 2015: (Start)
a(2*n,n) = A000108(n)*(2*n+1)^n;
a(3*n,2*n) = A001764(n)*(3*n+1)^(2*n);
a(p*n,(p-1)*n) = binomial(p*n,n)/((p-1)*n+1)*(p*n+1)^((p-1)*n) for p > 0;
Sum_{m=0..n} (m+1)*a(n,m) = (n+2)^n;
Sum_{m=0..n} (-1)^m*(m+1)*a(n,m) = (-n)^n where 0^0 = 1;
p(n,x) = Sum_{m=0..n} a(n,m)*x^m = ((1+(n+1)*x)^(n+1)-1)/((n+1)^2*x).
(End)

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A030524 A convolution triangle of numbers obtained from A036068.

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A033842 Triangle of coefficients of certain polynomials (exponents in decreasing order).

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A080421 a(n) = (n+1)*(n+2)*(n+9)*3^n/18.

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A053113 Expansion of (-1 + 1/(1-10*x)^10)/(100*x); related to A053109.

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A132166 A convolution triangle of numbers obtained from A036224.

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A049323 Triangle of coefficients of certain polynomials (exponents in increasing order), equivalent to A033842.

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A080421 a(n) = (n+1)(n+2)(n+9)*3^n/18.

A053113 Expansion of (-1 + 1/(1-10x)^10)/(100x); related to A053109.