A008277 -id:A008277 - cp's OEIS frontend

A069138 Triangle formed by multiplying Stirling numbers of 2nd kind S2(n,m) (A008277) by m+1.

2, 2, 3, 2, 9, 4, 2, 21, 24, 5, 2, 45, 100, 50, 6, 2, 93, 360, 325, 90, 7, 2, 189, 1204, 1750, 840, 147, 8, 2, 381, 3864, 8505, 6300, 1862, 224, 9, 2, 765, 12100, 38850, 41706, 18522, 3696, 324, 10, 2, 1533, 37320, 170525, 255150, 159789, 47040, 6750, 450, 11

Offset: 1

N. J. A. Sloane, Apr 10 2002

The number of rhyme schemes for a stanza of n+1 lines with m rhyming syllables in its first n lines.

			Triangle begins:
  2;
  2,  3;
  2,  9,   4;
  2, 21,  24,  5;
  2, 45, 100, 50, 6;
  ...

Suggested by R. K. Guy, Mar 11 2002.

Stephen Pollard, C.S. Peirce and the Bell Numbers, Mathematics Magazine, Vol. 76 (2003), pp. 99-106.

Row sums give Bell numbers A000110.

Cf. A360174 (Stirling1 counterpart), A360205 (Lah counterpart).

T(n, m) = stirling(n, m, 2)*(m+1);
tabl(nn) = for(n=1, nn, for (k=1, n, print1(T(n, m), ", ")); print); \\ Michel Marcus, Sep 21 2017

T(n, m) = (m+1)*S2(n, m).

More terms from Larry Reeves (larryr(AT)acm.org), Jul 01 2002

A088729 Matrix product of Stirling2-triangle A008277(n,k) and unsigned Lah-triangle |A008297(n,k)|.

Original entry on oeis.org

1, 3, 1, 13, 9, 1, 75, 79, 18, 1, 541, 765, 265, 30, 1, 4683, 8311, 3870, 665, 45, 1, 47293, 100989, 59101, 13650, 1400, 63, 1, 545835, 1362439, 960498, 278901, 38430, 2618, 84, 1, 7087261, 20246445, 16700545, 5844510, 1012431, 92610, 4494, 108, 1

Offset: 1

Vladeta Jovovic, Nov 22 2003

Also the Bell transform of A000670(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

Also the number of k-dimensional flats of the n-dimensional Catalan arrangement. - Shuhei Tsujie, May 05 2019

N. Nakashima and S. Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

Cf. A000670(first column), A075729(row sums).

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> add(combinat:-eulerian1(n+1, k)*2^k, k=0..n+1), 9); # Peter Luschny, Jan 26 2016

BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
B = BellMatrix[Function[n, HurwitzLerchPhi[1/2, -n-1, 0]/2], rows];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 27 2018, after Peter Luschny *)

E.g.f.: exp((exp(x)-1)*y/(2-exp(x))).

A088814 Matrix product of unsigned Lah-triangle |A008297(n,k)| and Stirling2-triangle A008277(n,k).

Original entry on oeis.org

1, 3, 1, 13, 9, 1, 73, 79, 18, 1, 501, 755, 265, 30, 1, 4051, 7981, 3840, 665, 45, 1, 37633, 93135, 57631, 13580, 1400, 63, 1, 394353, 1192591, 911582, 274141, 38290, 2618, 84, 1, 4596553, 16645431, 15285313, 5633922, 999831, 92358, 4494, 108, 1, 58941091

Offset: 1

Vladeta Jovovic, Nov 22 2003

Also the Bell transform of A000262(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

Cf. A000262(first column), A084357(row sums).

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> simplify(hypergeom([-n,-n-1],[],1)), 9); # Peter Luschny, Jan 26 2016

BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
B = BellMatrix[Function[n, Sum[BellY[n+1, k, Range[n+1]!], {k, 0, n+1}]], rows];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny_ *)

E.g.f.: exp(y*(exp(x/(1-x))-1)).

A227450 Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.

Original entry on oeis.org

2, 2, 4, 2, 12, 8, 2, 28, 48, 16, 2, 60, 200, 160, 32, 2, 124, 720, 1040, 480, 64, 2, 252, 2408, 5600, 4480, 1344, 128, 2, 508, 7728, 27216, 33600, 17024, 3584, 256, 2, 1020, 24200, 124320, 222432, 169344, 59136, 9216, 512, 2, 2044, 74640, 545680, 1360800, 1460928, 752640, 192000, 23040, 1024

Offset: 1

Geoffrey Critzer, Sep 22 2013

T(n,k) is the number of ways to separate {1,2,...,n} into 2 ordered subsets S,T so that the union of S and T = {1,2,...,n} then partition each subset so that the total number of blocks over both subsets is equal to k.
Triangle T(n,k), 1<=k<=n, read by rows, given by (0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...) DELTA (2, 0, 2, 0, 2, 0, 2, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 23 2013
Also the Bell transform of the constant sequence "a(n) = 2". For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016

			2,
2, 4,
2, 12, 8,
2, 28, 48, 16,
2, 60, 200, 160, 32,
2, 124, 720, 1040, 480, 64

Indranil Ghosh, Rows 1..125, flattened

Cf. A008277.

nn=8; a=Exp[x]-1; Map[Select[#, #>0&]&, Drop[Range[0,nn]! CoefficientList[Series[Exp[y a]^2, {x,0,nn}], {x,y}], 1]]//Grid
(* or *)
Flatten[Table[StirlingS2[n,k]*2^k,{n,1,10},{k,1,n}]] (* Indranil Ghosh, Feb 22 2017 *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[2&, rows = 12];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(lambda n: 2, 9); # Peter Luschny, Jan 29 2016

E.g.f.: A(x,y)^2 where A(x,y) is the e.g.f. for A008277.

A325503 Heinz number of row n of the triangle of Stirling numbers of the second kind A008277.

Original entry on oeis.org

2, 4, 20, 884, 528844, 3460086044, 340672148731996, 477782556719729075524, 11694209380474301218263758996, 4967476846044415922850025924897606724, 43298471669920632729336800855543564573041217668, 7790810575556906457316064931238939360882160372451591124244

Offset: 1

Gus Wiseman, May 07 2019

nonn

The Heinz number of a positive integer sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
                              2: {1}
                              4: {1,1}
                             20: {1,1,3}
                            884: {1,1,6,7}
                         528844: {1,1,10,15,25}
                     3460086044: {1,1,15,31,65,90}
                340672148731996: {1,1,21,63,140,301,350}
          477782556719729075524: {1,1,28,127,266,966,1050,1701}
  11694209380474301218263758996: {1,1,36,255,462,2646,3025,6951,7770}

Index entries for sequences related to Heinz numbers

Cf. A000040, A001222, A002870, A008277, A024412, A056239, A112798, A215366, A325500, A325502.

Times@@@Table[Prime[StirlingS2[n,k]],{n,1,10},{k,1,n}]

a(n) = Product_{i = 1..n} prime(A008277(n,i)).
A061395(a(n)) = A002870(n).
A056239(a(n)) = A000110(n).

A348649 Odd numbers in the triangle of Stirling numbers of the second kind (A008277).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 1, 1, 15, 25, 1, 1, 31, 65, 15, 1, 1, 63, 301, 21, 1, 1, 127, 1701, 1, 1, 255, 3025, 6951, 1, 1, 511, 34105, 42525, 22827, 45, 1, 1, 1023, 28501, 179487, 63987, 1155, 55, 1, 1, 2047, 611501, 159027, 22275, 1705, 1, 1, 4095, 261625, 7508501, 39325, 2431, 1

Offset: 1

Rémy Sigrist, Oct 27 2021

We take the odd values in A008277, as they appear, with duplicates.

For any n >= 1, the n-th row has A007306(n) terms.

			As an irregular table, the first rows are:
     1:    1;
     2:    1, 1;
     3:    1, 3, 1;
     4:    1, 7, 1;
     5:    1, 15, 25, 1;
     6:    1, 31, 65, 15, 1;
     7:    1, 63, 301, 21, 1;
     8:    1, 127, 1701, 1;
     9:    1, 255, 3025, 6951, 1;
    10:    1, 511, 34105, 42525, 22827, 45, 1;
    11:    1, 1023, 28501, 179487, 63987, 1155, 55, 1;
    ...

Rémy Sigrist, Table of n, a(n) for n = 1..10057 (rows for n = 1..331, flattened)
Rémy Sigrist, Logarithmic scatterplot of the first 35000 terms
Index entries for sequences related to Stirling numbers

See A014421, A014428, A014450, A014459 for similar sequences.

Cf. A007306, A008277, A348650 (even numbers).

row(n) = select(v -> v%2==1, vector(n, k, stirling(n, k, 2)))

A348650 Even numbers in the triangle of Stirling numbers of the second kind (A008277).

Original entry on oeis.org

6, 10, 90, 350, 140, 966, 1050, 266, 28, 7770, 2646, 462, 36, 9330, 5880, 750, 145750, 246730, 11880, 86526, 1379400, 1323652, 627396, 66, 2532530, 9321312, 5715424, 1899612, 359502, 78, 788970, 49329280, 20912320, 5135130, 752752, 66066, 42355950, 210766920

Offset: 4

Rémy Sigrist, Oct 27 2021

We take the even values in A008277, as they appear, with duplicates.

For any n >= 4, the n-th row has n - A007306(n) terms.

			As an irregular table, the first rows are:
     4:   6;
     5:   10;
     6:   90;
     7:   350, 140;
     8:   966, 1050, 266, 28;
     9:   7770, 2646, 462, 36;
    10:   9330, 5880, 750;
    11:   145750, 246730, 11880;
    12:   86526, 1379400, 1323652, 627396, 66;
    13:   2532530, 9321312, 5715424, 1899612, 359502, 78;
    14:   788970, 49329280, 20912320, 5135130, 752752, 66066;
    ...

Rémy Sigrist, Table of n, a(n) for n = 4..10120 (rows for n = 4..163, flattened)
Rémy Sigrist, Logarithmic scatterplot of the first 35000 terms
Index entries for sequences related to Stirling numbers

Cf. A007306, A008277, A348649 (odd numbers).

row(n) = select(v -> v%2==0, vector(n, k, stirling(n, k, 2)))

A091046 Stirling transform of first differences of Bell numbers (A005493), if offset zero: a(n) = Sum_{k=1..n} A008277(n,k)*A005493(k).

Original entry on oeis.org

1, 4, 20, 119, 817, 6338, 54707, 519184, 5366097, 59934937, 718748131, 9203953921, 125268224954, 1804750726306, 27426230051634, 438260834123607, 7343677070172330, 128716143768613600, 2354633702684629141, 44865189679858465163, 888784065003104357924

Offset: 1

Karol A. Penson, Dec 15 2003

nonn

Equals A039810 * [1,2,3,...], i.e., the square of the Stirling2 triangle and the natural number vector. - Gary W. Adamson, Jan 31 2008
From Mark Wildon, Nov 01 2022: (Start)
a(n) is the number of pairs (P, P') where P' is a set partition of {1,...,n}, P is a set partition of {1,...,P} refining P, and one part of P' is distinguished.
For example, for n=2 the 4 set partition pairs for n=2 are ({{1,2}},{{1,2}*}), ({{1},{2}},{{1,2}}*), ({{1},{2}},{{1}*,{2}}), ({{1},{2}},{{1},{2}*}), where the distinguished part of the coarser partition is marked *
a(n) is the inner product in the character ring of the symmetric group S_{mn} of the characters pi^n and phi_n Ind_{S_m wr S_n}^{S_{mn}}, where pi(g) = |Fix g| is the permutation character of the natural representation of S_{mn} and phi_n is the character of the wreath product S_m wr S_n obtained by inflating the character chi^{(n-1,1)} of S_n to S_m wr S_n. (End)

Alois P. Heinz, Table of n, a(n) for n = 1..478

Cf. A000258, A005493, A039810.

s:= proc(n) option remember; expand(`if`(n=0, 1,
      x*add(s(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
S:= proc(n, k) option remember; coeff(s(n), x, k) end:
b:= proc(n, k) option remember; `if`(k=0, n,
      add(S(n, j)*b(j, k-1), j=0..n))
    end:
a:= n-> b(n, 2):
seq(a(n), n=1..23);  # Alois P. Heinz, Aug 24 2021

len = 23;
Array[StirlingS2, {len, len}].Differences[Array[BellB, len+1]] (* Jean-François Alcover, Apr 25 2022 *)

E.g.f.: (exp(exp(x)-1)-1)*exp(exp(exp(x)-1)-1).

Representation as an infinite sum (Dobinski-type relation): a(n) = exp(exp(-1)-1)*Sum(p^n*((Sum((Stirling2(p+1, k) - Stirling2(p, k))*exp(-k), k=1..p) + exp(-(p+1)))/p!), p>=1), n = 1, 2, ....

A112339 Triangle read by rows of numbers b_{n,k}, n >= 2, 1 <= k < n such that (1/(1-qt))Product_{n,k} 1/(1 - q^nt^k)^b_{n,k} = Sum_{i,j>=1} S_{i,j} q^it^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).

Original entry on oeis.org

1, 1, 2, 1, 5, 3, 1, 13, 16, 4, 1, 28, 67, 34, 5, 1, 60, 249, 229, 65, 6, 1, 123, 853, 1265, 609, 107, 7, 1, 251, 2787, 6325, 4696, 1376, 168, 8, 1, 506, 8840, 29484, 31947, 14068, 2772, 244, 9, 1, 1018, 27503, 131402, 199766, 124859, 36252, 5118, 345, 10

Offset: 2

Mike Zabrocki, Sep 05 2005

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

			Triangle begins:
  1;
  1,  2;
  1,  5,  3;
  1, 13, 16,  4;
  ...

Cf. A008277, A085686, A112340.

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..11)]);

A137649 Triangle read by rows, A000012 * A008277.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 11, 7, 1, 5, 26, 32, 11, 1, 6, 57, 122, 76, 16, 1, 7, 120, 423, 426, 156, 22, 1, 8, 247, 1389, 2127, 1206, 288, 29, 1, 9, 502, 4414, 9897, 8157, 2934, 491, 37, 1, 10, 1013, 13744, 44002, 50682, 25761, 6371, 787, 46, 1

Offset: 1

Gary W. Adamson, Feb 01 2008

Row sums = A024716: (1, 3, 8, 23, 75, 278, ...).

A137650 = A008277 * A000012.

			First few rows of the triangle:
  1;
  2,   1;
  3,   4,   1;
  4,  11,   7,   1;
  5,  26,  32,  11,   1;
  6,  57, 122,  76,  16,  1;
  7, 120, 423, 426, 156, 22, 1;
  ...
Row 4 = (4, 11, 7, 1) = partial column sums of the first 4 rows of A008277:
  1;
  1, 1;
  1, 3, 1;
  1, 7, 6, 1;
  ...

Cf. A024716, A008277, A137650.

A000012 * A008277 (Stirling2 triangle) as infinite lower triangular matrices. Partial column sums of A008277.

cp's OEIS Frontend

A069138 Triangle formed by multiplying Stirling numbers of 2nd kind S2(n,m) (A008277) by m+1.

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A088729 Matrix product of Stirling2-triangle A008277(n,k) and unsigned Lah-triangle |A008297(n,k)|.

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A088814 Matrix product of unsigned Lah-triangle |A008297(n,k)| and Stirling2-triangle A008277(n,k).

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A227450 Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.

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A325503 Heinz number of row n of the triangle of Stirling numbers of the second kind A008277.

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A348649 Odd numbers in the triangle of Stirling numbers of the second kind (A008277).

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A348650 Even numbers in the triangle of Stirling numbers of the second kind (A008277).

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A091046 Stirling transform of first differences of Bell numbers (A005493), if offset zero: a(n) = Sum_{k=1..n} A008277(n,k)*A005493(k).

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A112339 Triangle read by rows of numbers b_{n,k}, n >= 2, 1 <= k < n such that (1/(1-qt))Product_{n,k} 1/(1 - q^nt^k)^b_{n,k} = Sum_{i,j>=1} S_{i,j} q^it^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).