A008277 -id:A008277 - cp's OEIS frontend

A137854 Triangle generated from an array: A008277 * A008277(transform).

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 11, 8, 1, 1, 16, 28, 28, 16, 1, 1, 32, 71, 87, 71, 32, 1, 1, 64, 184, 266, 266, 184, 64, 1, 1, 128, 491, 823, 952, 823, 491, 128, 1, 1, 256, 1348, 2598, 3381, 381, 2598, 1348, 2561

Offset: 1

Gary W. Adamson, Feb 15 2008

Row sums = A000995 such that row 1 = A000995(3) = 1.
This array is the product of the lower triangular Stirling matrix and its transpose, which explains why the array is symmetric. - David Callan, Dec 02 2011
In the triangle, T(n,k) is the number of permutations of [n+1] that avoid both dashed patterns 1-23 and 3-12, start with an ascent, and have first entry k. For example, T(4,2)=4 counts 23154, 24153, 24315, 25431. - David Callan, Dec 02 2011

			First few rows of the array:
  1,  1,  1,   1,   1,    1, ...
  1,  2,  4,   8,  16,   32, ...
  1,  4, 11,  28,  71,  184, ...
  1,  8, 28,  87, 266,  823, ...
  1, 16, 71, 266, 952, 3381, ...
  ...
First few rows of the triangle:
  1;
  1,   1;
  1,   2,   1;
  1,   4,   4,   1;
  1,   8,  11,   8,   1;
  1,  16,  28,  28,  16,   1;
  1,  32,  71,  87,  71,  32,   1;
  1,  64, 184, 266, 266, 184,  64,   1;
  1, 128, 491, 823, 952, 823, 491, 128,   1;
  ...

Cf. A000995, A008277.

Triangle read by rows = antidiagonals of an array formed by A008277 * A008277(transform), where A008277 = the Stirling number of the second kind triangle.

A154959 Triangle read by rows. Signed version of A008277.

Original entry on oeis.org

1, -1, 1, -1, -3, 1, -1, -7, -6, 1, -1, -15, -25, -10, 1, -1, -31, -90, -65, -15, 1, -1, -63, -301, -350, -140, -21, 1, -1, -127, -966, -1701, -1050, -266, -28, 1, -1, -255, -3025, -7770, -6951, -2646, -462, -36, 1

Offset: 1

Mats Granvik, Jan 18 2009

Main diagonal positive, the rest of the terms negative. Matrix inverse of this triangle is A154960. Signs in columns as in A153881.

A223512 Triangle T(n,k) represents the coefficients of (x^10*d/dx)^n, where n=1,2,3,...;generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

Original entry on oeis.org

1, 10, 1, 190, 30, 1, 5320, 1060, 60, 1, 196840, 45600, 3400, 100, 1, 9054640, 2340040, 208800, 8300, 150, 1, 498005200, 140096880, 14241640, 690200, 17150, 210, 1, 31872332800, 9604302400, 1080045120, 60485040, 1856400, 31640, 280, 1, 2326680294400

Offset: 1

Udita Katugampola, Mar 23 2013

			1;
10,1;
190,30,1;
5320,1060,60,1;
196840,45600,3400,100,1;
9054640,2340040,208800,8300,150,1;
498005200,140096880,14241640,690200,17150,210,1;
31872332800,9604302400,1080045120,60485040,1856400,31640,280,1,2326680294400

Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522, A223168-A223172, A223523-A223532.

b[0]:=g(x):
for j from 1 to 10 do
b[j]:=simplify(x^10*diff(b[j-1],x$1);
end do;

A381160 a(n) is the permanent of the n X n matrix whose element (i,j) is equal to A008277(i+3, j) with 1 <= i,j <= n.

Original entry on oeis.org

1, 1, 22, 3206, 1902936, 3504528354, 16660734321540, 179059038168086056, 3938830136216956996632, 164125096331945477980176920, 12173562237817299484378342192768, 1527294306324982018922212102518520032, 310564445230567070838152555220146533261496, 98712056006032672983172826864304778359411112064

Offset: 0

Stefano Spezia, Feb 15 2025

nonn

			a(3) = 3206:
  [1,  7,  6]
  [1, 15, 25]
  [1, 31, 90]

Cf. A000442 (determinant), A008277, A381166.

a[n_]:=Permanent[Table[StirlingS2[i+3,j],{i,n},{j,n}]]; Join[{1},Array[a,13]]

a(n) = matpermanent(matrix(n, n, i, j, stirling(i+3,j,2))); \\ Michel Marcus, Feb 16 2025

A381166 a(n) is the permanent of the n X n matrix whose element (i,j) is equal to A008277(i+4, j) with 1 <= i,j <= n.

Original entry on oeis.org

1, 1, 46, 23216, 70437736, 911400637082, 39931366088759328, 5015203546888139970264, 1592320463242701429692077472, 1158339311156769223634640734447744, 1783702957209729441902140461938160455424, 5447268928199100257603373050876725987854119216, 31237114830378466799129128930824084710690680271414364

Offset: 0

Stefano Spezia, Feb 15 2025

nonn

			a(3) = 23216:
  [1, 15,  25]
  [1, 31,  90]
  [1, 63, 301]

Cf. A008277, A134375 (determinant), A381160.

a[n_]:=Permanent[Table[StirlingS2[i+4, j], {i, n}, {j, n}]]; Join[{1}, Array[a, 12]]

a(n) = matpermanent(matrix(n, n, i, j, stirling(i+4,j,2))); \\ Michel Marcus, Feb 16 2025

A166280 Stirling2 triangle mod 2, T(n,k) = A008277(n,k) mod 2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1

Offset: 0

Gerald McGarvey, Oct 10 2009

			Triangle begins:
1,
1,1,
1,1,1,
1,1,0,1,
1,1,1,0,1,
1,1,0,1,1,1,
1,1,1,0,0,1,1,
1,1,0,1,0,0,0,1,
1,1,1,0,1,0,0,0,1,
1,1,0,1,1,1,0,0,1,1,
1,1,1,0,0,1,1,0,1,1,1,
1,1,0,1,0,0,0,1,1,1,0,1,
1,1,1,0,1,0,0,0,0,1,1,0,1,
...

Cf. A008277, A047999 (Sierpinski's triangle, Pascal's triangle mod 2).

p = 2; s=14; S2T = matrix(s,s,n,k, if(k==1,1)); for(n=2,s,for(k=2,n, S2T[n,k]=k*S2T[n-1,k]+S2T[n-1,k-1]));
S2TMP = matrix(s,s,n,k, S2T[n,k]%p);
for(n=1,s,for(k=1,n,print1(S2TMP[n,k]," "));print())

A187556 Triangle read by rows of products of (signless) Stirling numbers of the first kind (A132393) and Stirling numbers of the second kind (A008277).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 9, 1, 0, 6, 77, 36, 1, 0, 24, 750, 875, 100, 1, 0, 120, 8494, 20250, 5525, 225, 1, 0, 720, 111132, 488824, 257250, 24500, 441, 1, 0, 5040, 1659636, 12685512, 11514069, 2058000, 85652, 784, 1, 0, 40320, 27943920, 357325100, 522796680, 156042999, 12002256, 252252, 1296, 1, 0, 362880, 524580336, 10941291000, 24681106400, 11453045625, 1444332771, 55566000, 652500, 2025, 1

Offset: 0

Emanuele Munarini, Mar 11 2011

			Triangle begins:
1
0,1
0,1,1
0,2,9,1
0,6,77,36,1
0,24,750,875,100,1
0,120,8494,20250,5525,225,1
0,720,111132,488824,257250,24500,441,1
0,5040,1659636,12685512,11514069,2058000,85652,784,1

Cf. A132393, A008277

seq(seq(abs(combinat[stirling1](n,k))*combinat[stirling2](n,k),k=0..n),n=0..8);

Flatten[Table[Table[Abs[StirlingS1[n, k]]*StirlingS2[n, k], {k, 0, n}],{n, 0, 8}] ,1]

create_list(abs(stirling1(n,k)*stirling2(n,k)),n,0,10,k,0,n);

Formula: a(n,k) = s(n,k)*S(n,k), where the s(n,k) are the (signless) Stirling numbers of the first kind and the S(n,k) are the Stirling numbers of the second kind.

A187557 Triangle read by rows of products of Stirling numbers of the second kind (A008277): a(n,k) = S(n,k) S(n+1,k+1).

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 7, 18, 1, 0, 15, 175, 60, 1, 0, 31, 1350, 1625, 150, 1, 0, 63, 9331, 31500, 9100, 315, 1, 0, 127, 60858, 512001, 367500, 37240, 588, 1, 0, 255, 384175, 7505820, 11823651, 2778300, 122892, 1008, 1, 0, 511, 2379150, 103167625, 330419250, 158670477, 15558480, 346500, 1620, 1, 0, 1023, 14564011, 1359847500, 8414726650, 7632684675, 1460631249, 69854400, 866250, 2475, 1

Offset: 0

Emanuele Munarini, Mar 11 2011

			Triangle begins:
1
0,1
0,3,1
0,7,18,1
0,15,175,60,1
0,31,1350,1625,150,1
0,63,9331,31500,9100,315,1
0,127,60858,512001,367500,37240,588,1
0,255,384175,7505820,11823651,2778300,122892,1008,1

Cf. A008277

seq(seq(combinat[stirling2](n,k)*combinat[stirling2](n+1,k+1),k=0..n),n=0..8);

Table[StirlingS2[n, k]StirlingS2[n + 1, k + 1], {n, 0, 8}, {k, 0, 8}]//MatrixForm

create_list(stirling2(n,k)*stirling2(n+1,k+1),n,0,10,k,0,n);

A223514 Triangle T(n,k) represents the coefficients of (x^12*d/dx)^n, where n=1,2,3,...; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

Original entry on oeis.org

1, 12, 1, 276, 36, 1, 9384, 1536, 72, 1, 422280, 80040, 4920, 120, 1, 23647680, 4984560, 365400, 12000, 180, 1, 1584394560, 362597760, 30197160, 1205400, 24780, 252, 1, 123582775680, 30229617600, 2778370560, 127834560, 3237360, 45696, 336, 1, 1099867035520

Offset: 1

Udita Katugampola, Mar 23 2013

			1;
12,1;
276,36,1;
9384,1536,72,1;
422280,80040,4920,120,1;
23647680,4984560,365400,12000,180,1;
1584394560,362597760,30197160,1205400,24780,252,1;
123582775680,30229617600,2778370560,127834560,3237360,45696,336,1;
1099867035520,...

Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522, A223168-A223172, A223523-A223532.

b[0]:=f(x):
for j from 1 to 10 do
b[j]:=simplify(x^12*diff(b[j-1],x$1);
end do;

A223515 Triangle T(n,k) represents the coefficients of (x^13*d/dx)^n, where n=1,2,3,...; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

Original entry on oeis.org

1, 13, 1, 325, 39, 1, 12025, 1807, 78, 1, 589225, 102375, 5785, 130, 1, 35942725, 6936475, 466830, 14105, 195, 1, 2623818925, 549241875, 41948725, 1538810, 29120, 273, 1, 223024608625, 49858620175, 4198780950, 177364005, 4130490, 53690, 364, 1, 21633387036625

Offset: 1

Udita Katugampola, Mar 23 2013

			1;
13,1;
325,39,1;
12025,1807,78,1;
589225,102375,5785,130,1;
35942725,6936475,466830,14105,195,1
2623818925,549241875,41948725,1538810,29120,273,1;
223024608625,49858620175,4198780950,177364005,4130490,53690,364,1;
21633387036625,...

Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522, A223168-A223172, A223523-A223532.

b[0]:=f(x):
for j from 1 to 10 do
b[j]:=simplify(x^13*diff(b[j-1],x$1);
end do;

cp's OEIS Frontend

A137854 Triangle generated from an array: A008277 * A008277(transform).

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A154959 Triangle read by rows. Signed version of A008277.

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A223512 Triangle T(n,k) represents the coefficients of (x^10*d/dx)^n, where n=1,2,3,...;generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

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Maple

A381160 a(n) is the permanent of the n X n matrix whose element (i,j) is equal to A008277(i+3, j) with 1 <= i,j <= n.

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PARI

A381166 a(n) is the permanent of the n X n matrix whose element (i,j) is equal to A008277(i+4, j) with 1 <= i,j <= n.

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Mathematica

PARI

A166280 Stirling2 triangle mod 2, T(n,k) = A008277(n,k) mod 2.

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PARI

A187556 Triangle read by rows of products of (signless) Stirling numbers of the first kind (A132393) and Stirling numbers of the second kind (A008277).

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Formula

A187557 Triangle read by rows of products of Stirling numbers of the second kind (A008277): a(n,k) = S(n,k) S(n+1,k+1).

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Maple

Mathematica

Maxima

A223514 Triangle T(n,k) represents the coefficients of (x^12*d/dx)^n, where n=1,2,3,...; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

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Maple

A223515 Triangle T(n,k) represents the coefficients of (x^13*d/dx)^n, where n=1,2,3,...; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

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