A008277 -id:A008277 - cp's OEIS frontend

A112340 Triangle read by rows of numbers b_{n,k}, n>=1, 1<=k<=n such that Product_{n,k} 1/(1-q^n t^k)^{b_{n,k}} = 1 + Sum_{i,j>=1} S_{i,j} q^i t^j where S_{i,j} are entries in the table A008277 (the inverse Euler transformation of the table of Stirling numbers of the second kind).

1, 1, 0, 1, 2, 0, 1, 5, 3, 0, 1, 13, 16, 4, 0, 1, 28, 67, 34, 5, 0, 1, 60, 249, 229, 65, 6, 0, 1, 123, 853, 1265, 609, 107, 7, 0, 1, 251, 2787, 6325, 4696, 1376, 168, 8, 0, 1, 506, 8840, 29484, 31947, 14068, 2772, 244, 9, 0, 1, 1018, 27503, 131402, 199766, 124859, 36252

Offset: 1

Mike Zabrocki, Sep 05 2005; Aug 06 2006

Row sums equal to A085686, second column = A084174 - 1

The number of set partitions of size n length k which are 'Lyndon,' that is, since all set partitions are isomorphic to sequences of atomic set partitions (A087903), those which are smallest of all rotations of these sequences in lex order (with respect to some ordering on the atomic set partitions) are Lyndon. 1; 1, 0; 1, 2, 0; 1, 5, 3, 0; 1, 13, 16, 4, 0;

			There are 6 set partitions of size 4 and length 3, {12|3|4}, {13|2|4}, {14|2|3}, {1|23|4}, {1|24|3}, {1|2|34} and the sequences the correspond to are ({12},{1},{1}), ({13|2}, {1}), ({14|2|3}), ({1},{12},{1}), ({1},{13|2}), ({1},{1},{12}). Now there are three {({12},{1},{1}), ({1},{12},{1}), ({1},{1},{12})} that are rotations of each other and ({1}, {1}, {12}) is the smallest of these, {({13|2}, {1}), ({1},{13|2})} are rotations of each other and ({1},{13|2}) is the smallest and ({14|2|3}) is atomic and all atomic s.p. are Lyndon. Hence {1|2|34}, {1|24|3}, {14|2|3} are Lyndon and a(4,3) = 3
Triangle begins:
  1;
  1,  0;
  1,  2,  0;
  1,  5,  3,  0;
  1, 13, 16,  4, 0;
  1, 28, 67, 34, 5, 0;
  ...

N. Bergeron, M. Zabrocki, The Hopf algebras of symmetric functions and quasisymmetric functions in non-commutative variables are free and cofree, arXiv:math/0509265 [math.CO], 2005.
M. Rosas and B. Sagan, Symmetric Functions in Noncommuting Variables, Transactions of the American Mathematical Society, 358 (2006), no. 1, 215-232.
M. C. Wolf, Symmetric functions for non-commutative elements, Duke Math. J., 2 (1936), 626-637.

Cf. A008277, A085686, A112339.

Cf. A087903, A000110.

EULERitable:=proc(tbl) local ser,out,i,j,tmp; ser:=1+add(add(q^i*t^j*tbl[i][j], j=1..nops(tbl[i])), i=1..nops(tbl)); out:=[]; for i from 1 to nops(tbl) do tmp:=coeff(ser,q,i); ser:=expand(ser*mul(add((-q^i*t^j)^k*choose(abs(coeff(tmp,t,j)),k),k=0..nops(tbl)/i), j = 1..degree(tmp,t))); ser:=subs({seq(q^j=0,j=nops(tbl)+1..degree(ser,q))},ser); out:=[op(out),[seq(abs(coeff(tmp,t,j)), j=1..degree(tmp,t))]]; end do; out; end: EULERitable([seq([seq(combinat[stirling2](n,k),k=1..n)],n=1..10)]);

nmax = 11; b[n_, k_] /; k < 1 || k > n = 0;
coes[m_] := Product[1/(1 - q^n t^k)^b[n, k], {n, 1, m}, {k, 1, m}] - 1 - Sum[ StirlingS2[i, j] q^i t^j, {i, 1, m}, {j, 1, m}] + O[t]^m + O[q]^m // Normal // CoefficientList[#, {t, q}]&;
sol[1] = {b[1, 1] -> 1};
Do[sol[m] = Solve[Thread[(coes[m] /. sol[m - 1]) == 0]], {m, 2, nmax + 1}];
bb = Flatten[Table[sol[m], {m, 1, nmax + 1}]];
Table[b[n, k] /. bb, {n, 1, nmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 11 2017 *)

A137650 Triangle read by rows, A008277 * A000012.

Original entry on oeis.org

1, 2, 1, 5, 4, 1, 15, 14, 7, 1, 52, 51, 36, 11, 1, 203, 202, 171, 81, 16, 1, 877, 876, 813, 512, 162, 22, 1, 4140, 4139, 4012, 3046, 1345, 295, 29, 1, 21147, 21146, 20891, 17866, 10096, 3145, 499, 37, 1, 115975, 115974, 115463

Offset: 1

Gary W. Adamson, Feb 01 2008

Left column = Bell numbers (A000110) starting (1, 2, 5, 15, 52, 203, ...). Row sums = A005493(n+1): (1, 3, 10, 37, 151, 674, ...).

Corresponding to the generalized Stirling number triangle of first kind A049444. - Peter Luschny, Sep 18 2011

			First few rows of the triangle are
    1;
    2,   1;
    5,   4,   1;
   15,  14,   7,   1;
   52,  51,  36,  11,   1;
  203, 202, 171,  81,  16,   1;
  877, 876, 813, 512, 162,  22,   1;
  ...

Cf. A000110, A008277, A005493, A049444.

A similar triangle is A133611.

A137650_row := proc(n) local k,i;
add(add(combinat[stirling2](n, n-i), i=0..k)*x^(n-k-1),k=0..n-1);
seq(coeff(%,x,k),k=0..n-1) end:
seq(print(A137650_row(n)),n=1..7); # Peter Luschny, Sep 18 2011

row[n_] := Table[StirlingS2[n, k], {k, 0, n}] // Reverse // Accumulate // Reverse // Rest;
Array[row, 10] // Flatten (* Jean-François Alcover, Dec 07 2019 *)

A008277 * A000012 as infinite lower triangular matrices. Partial sums of A008277 rows starting from the right.

A039812 Triangle read by rows: matrix 4th power of the Stirling2 triangle A008277.

Original entry on oeis.org

1, 4, 1, 22, 12, 1, 154, 136, 24, 1, 1304, 1650, 460, 40, 1, 12915, 21904, 8550, 1160, 60, 1, 146115, 318521, 162904, 30590, 2450, 84, 1, 1855570, 5051988, 3246068, 789824, 86940, 4592, 112, 1, 26097835, 86910426, 68151304, 20606796, 2919504, 210924, 7896, 144, 1

Offset: 1

Christian G. Bower, Feb 15 1999

			Triangle begins
      1;
      4,     1;
     22,    12,    1;
    154,   136,   24,    1;
   1304,  1650,  460,   40,  1;
  12915, 21904, 8550, 1160, 60, 1;
  ...

Seiichi Manyama, Rows n = 1..140, flattened

Cf. A008277, A000307 (first column).

Cf. A039810, A039811, A039813.

Flatten[Table[SeriesCoefficient[(Exp[Exp[Exp[Exp[x]-1]-1]-1]-1)^k, {x,0,n}]  n!/k!, {n,9}, {k,n}]] (* Stefano Spezia, Sep 12 2022 *)

E.g.f. k-th column: (( exp(exp(exp(exp(x)-1)-1)-1)-1 )^k)/k!. [corrected by Seiichi Manyama, Feb 12 2022]

A049434 Stirling numbers of second kind: 8th column of Stirling2 triangle A008277.

Original entry on oeis.org

1, 36, 750, 11880, 159027, 1899612, 20912320, 216627840, 2141764053, 20415995028, 189036065010, 1709751003480, 15170932662679, 132511015347084, 1142399079991620, 9741955019900400, 82318282158320505, 690223721118368580, 5749622251945664950

Offset: 8

Wolfdieter Lang

See A000771.

Cf. A000225, A000392, A000453, A000481, A000770, A000771, A049435. a(n)= A008277(n, 8).

lst={};Do[f=StirlingS2[n, 8];AppendTo[lst, f], {n, 8, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)
CoefficientList[Series[1/((1 - x) (1 - 2 x) (1 - 3 x) (1 - 4 x) (1 - 5 x) (1 - 6 x) (1 - 7 x) (1 - 8 x)), {x, 0, 25}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)

G.f.: x^8/product_{k=1..8} (1-k*x).
E.g.f.: ((exp(x)-1)^8)/8!.
a(n) = det(|s(i+8,j+7)|, 1 <= i,j <= n-8), where s(n,k) are Stirling numbers of the first kind. - Mircea Merca, Apr 06 2013

A049435 Stirling numbers of second kind: 10th column of Stirling2 triangle A008277.

Original entry on oeis.org

1, 55, 1705, 39325, 752752, 12662650, 193754990, 2758334150, 37112163803, 477297033785, 5917584964655, 71187132291275, 835143799377954, 9593401297313460, 108254081784931500, 1203163392175387500, 13199555372846848005, 143197070509423605675

Offset: 10

Wolfdieter Lang

See A000771.

Index entries for linear recurrences with constant coefficients, signature (55,-1320,18150,-157773,902055,-3416930,8409500,-12753576,10628640,-3628800).

Cf. A000225, A000392, A000453, A000481, A000770, A000771, A049434, A049447. a(n) = A008277(n, 10).

lst={};Do[f=StirlingS2[n, 10];AppendTo[lst, f], {n, 10, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)
CoefficientList[Series[1/((1 - x) (1 - 2 x) (1 - 3 x) (1 - 4 x) (1 - 5 x) (1 - 6 x) (1 - 7 x) (1 - 8 x) (1 - 9 x) (1 - 10 x)), {x, 0, 25}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)
StirlingS2[Range[10,35],10] (* Harvey P. Dale, Nov 07 2020 *)

G.f.: x^10/Product_{k=1..10} (1-k*x).
E.g.f.: ((exp(x)-1)^10)/10!.
a(n) = det(|s(i+10,j+9)|, 1 <= i,j <= n-10), where s(n,k) are Stirling numbers of the first kind. - Mircea Merca, Apr 06 2013

A049447 Stirling numbers of second kind: 9th column of Stirling2 triangle A008277.

Original entry on oeis.org

1, 45, 1155, 22275, 359502, 5135130, 67128490, 820784250, 9528822303, 106175395755, 1144614626805, 12011282644725, 123272476465204, 1241963303533920, 12320068811796900, 120622574326072500, 1167921451092973005, 11201516780955125625, 106563273280541795575

Offset: 9

Wolfdieter Lang

See A000771.

Cf. A000225, A000392, A000453, A000481, A000770, A000771, A049434.

lst={};Do[f=StirlingS2[n, 9];AppendTo[lst, f], {n, 9, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)
CoefficientList[Series[1/((1 - x) (1 - 2 x) (1 - 3 x) (1 - 4 x) (1 - 5 x) (1 - 6 x) (1 - 7 x) (1 - 8 x) (1 - 9 x)), {x, 0, 25}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)
StirlingS2[Range[9,30],9] (* Harvey P. Dale, Dec 12 2022 *)

a(n)= A008277(n, 9).
G.f.: x^9/product_{k=1..9} (1-k*x).
E.g.f.: ((exp(x)-1)^9)/9!.
a(n) = det(|s(i+9,j+8)|, 1 <= i,j <= n-9), where s(n,k) are Stirling numbers of the first kind. - Mircea Merca, Apr 06 2013

A137597 Triangle read by rows: A008277 * A007318.

Original entry on oeis.org

1, 2, 1, 5, 5, 1, 15, 22, 9, 1, 52, 99, 61, 14, 1, 203, 471, 385, 135, 20, 1, 877, 2386, 2416, 1140, 260, 27, 1, 4140, 12867, 15470, 9156, 2835, 455, 35, 1, 21147, 73681, 102215, 72590, 28441, 6230, 742, 44, 1

Offset: 1

Gary W. Adamson, Jan 29 2008

Row sums = A035009 starting (1, 3, 11, 47, 227, ...).

			First few rows of the triangle:
    1;
    2,   1;
    5,   5,   1;
   15,  22,   9,   1;
   52,  99,  61,  14,  1;
  203, 471, 385, 135, 20, 1;
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows n = 1..150, flattened)
Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See p. 16.

Cf. A035009, A008277.

Cf. A126350.

T:= (n, k)-> add(Stirling2(n, j)*binomial(j-1, k-1), j=k..n):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Sep 03 2019

Table[Sum[StirlingS2[n, j]*Binomial[j - 1, k - 1], {j, k, n}], {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Aug 31 2023 *)

A008277 * A007318 as infinite lower triangular matrices.

A274712 a(n) = A008277(3n-1,n) / (n(n+1)/2) for n>=1, where A008277 are the Stirling numbers of the second kind.

Original entry on oeis.org

1, 5, 161, 14575, 2671669, 833607138, 397984073059, 270609861663900, 248922595132336125, 298037055910658382175, 450755158919281716609746, 840770855566250627155136090, 1896671776639253430025972662743, 5091278095597325836977485757711800, 16040729445423172146341201903726496024, 58625927208516621021861960954787323034320, 246047331971247756894582227572712664877434765, 1175344062721738572130662103242054758238706829325

Offset: 1

Paul D. Hanna, Jul 03 2016

nonn

Paul D. Hanna, Table of n, a(n) for n = 1..100

Cf. A274713.

{a(n) = polcoeff( 1/prod(k=1, n, 1-k*x +x*O(x^(2*n))), 2*n-1) / (n*(n+1)/2)}
for(n=1, 20, print1(a(n), ", "))

{a(n) = abs( stirling(3*n-1, n, 2) / (n*(n+1)/2) )}
for(n=1, 20, print1(a(n), ", "))

{a(n) = 1/n! * sum(k=0, n, (-1)^(n-k) * binomial(n, k) * k^(3*n-1)) / (n*(n+1)/2)}
for(n=1, 20, print1(a(n), ", "))

a(n) = A274713(n) / (n*(n+1)/2), where A274713(n) is the number of partitions of a {3*n-1}-set into n nonempty subsets.
a(n) = 1/n! * Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * k^(3*n-1) / (n*(n+1)/2).
a(n) ~ sqrt(2) * 3^(3*n-1) * n^(2*n-7/2) / (exp(2*n) * c^n * (3-c)^(2*n-1) * sqrt(Pi*(1-c))), where c = -LambertW(-3*exp(-3)) = 0.1785606278779211... = -A226750. - Vaclav Kotesovec, Jul 06 2016

A055896 Exponential transform of Stirling2 triangle A008277.

Original entry on oeis.org

1, 1, 2, 1, 6, 5, 1, 14, 30, 15, 1, 30, 125, 150, 52, 1, 62, 450, 975, 780, 203, 1, 126, 1505, 5250, 7280, 4263, 877, 1, 254, 4830, 25515, 54600, 53998, 24556, 4140, 1, 510, 15125, 116550, 361452, 537138, 405174, 149040, 21147, 1, 1022, 46650

Offset: 1

Christian G. Bower, Jun 09 2000

			Triangle begins
  1;
  1,   2;
  1,   6,   5;
  1,  14,  30,  15;
  1,  30, 125, 150,  52; ...

N. J. A. Sloane, Transforms

Row sums give A000258.

nn=8; a=Exp[x]-1; Drop[Map[Select[#, #>0&]&, Range[0,nn]! CoefficientList[Series[Exp[Exp[y a]-1], {x,0,nn}], {x,y}]], 1]//Grid (* Geoffrey Critzer, Sep 22 2013 *)

E.g.f.: A(x, y) = exp(exp(y*exp(x)-y)-1).

A061113 Concatenation of numbers in n-th row of triangle of Stirling numbers of second kind (A008277).

Original entry on oeis.org

1, 11, 131, 1761, 11525101, 1319065151, 163301350140211, 112796617011050266281, 12553025777069512646462361, 151193303410542525228275880750451

Offset: 1

Amarnath Murthy, Apr 21 2001

Amarnath Murthy, A general result on the Smarandache Star function, Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000.
Amarnath Murthy, Properties of Smarandache Star Triangle, Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000.

with(combinat, stirling2): for n from 1 to 15 do for k from 1 to n do printf(`%d`, stirling2(n,k)) od: printf(`,`): od:

More terms from James Sellers, Apr 23 2001

cp's OEIS Frontend

A112340 Triangle read by rows of numbers b_{n,k}, n>=1, 1<=k<=n such that Product_{n,k} 1/(1-q^n t^k)^{b_{n,k}} = 1 + Sum_{i,j>=1} S_{i,j} q^i t^j where S_{i,j} are entries in the table A008277 (the inverse Euler transformation of the table of Stirling numbers of the second kind).

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A137650 Triangle read by rows, A008277 * A000012.

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A039812 Triangle read by rows: matrix 4th power of the Stirling2 triangle A008277.

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A049434 Stirling numbers of second kind: 8th column of Stirling2 triangle A008277.

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A049435 Stirling numbers of second kind: 10th column of Stirling2 triangle A008277.

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A049447 Stirling numbers of second kind: 9th column of Stirling2 triangle A008277.

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A137597 Triangle read by rows: A008277 * A007318.

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A274712 a(n) = A008277(3*n-1,n) / (n*(n+1)/2) for n>=1, where A008277 are the Stirling numbers of the second kind.

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A274712 a(n) = A008277(3n-1,n) / (n(n+1)/2) for n>=1, where A008277 are the Stirling numbers of the second kind.