A049376 -id:A049376 - cp's OEIS frontend

A046089 Triangle read by rows, the Bell transform of (n+2)!/2 without column 0.

1, 3, 1, 12, 9, 1, 60, 75, 18, 1, 360, 660, 255, 30, 1, 2520, 6300, 3465, 645, 45, 1, 20160, 65520, 47880, 12495, 1365, 63, 1, 181440, 740880, 687960, 235305, 35700, 2562, 84, 1, 1814400, 9072000, 10372320, 4452840, 877905, 86940, 4410, 108, 1

Offset: 1

Wolfdieter Lang

Previous name was: A triangle of numbers related to triangle A030523.
a(n,1)= A001710(n+1). a(n,m)=: S1p(3; n,m), a member of a sequence of lower triangular Jabotinsky matrices with nonnegative entries, including S1p(1; n,m)= A008275 (unsigned Stirling first kind), S1p(2; n,m)= A008297(n,m) (unsigned Lah numbers).
Signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A035342(n,m) := S2(3; n,m). The monic row polynomials E(n,x) := sum(a(n,m)*x^m,m=1..n), E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
a(n,m) enumerates unordered increasing n-vertex m-forests composed of m unary trees (out-degree r from {0,1}) whose vertices of depth (distance from the root) j>=1 come in j+2 colors. The k roots (j=0) each come in one (or no) color. - Wolfdieter Lang, Oct 12 2007
a(4,2)=75=4*(3*4)+3*(3*3) from the two types of unordered 2-forests of unary increasing trees associated with the two m=2 parts partitions (1,3) and (2^2) of n=4. The first type has 4 increasing labelings, each coming in (1)*(1*3*4)=12 colored versions, e.g. ((1c1),(2c1,3c3,4c2)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 3 colors, c1, c2 and c3, can be chosen and the vertex labeled 4 with j=2 can come in 4 colors, e.g. c1, c2, c3 and c4. Therefore there are 4*(1)*(1*3*4)=48 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*3)*(1*3))=27 such forests, e.g. ((1c1,3c2)(2c1,4c1)) or ((1c1,3c2)(2c1,4c2)), etc. - Wolfdieter Lang, Oct 12 2007
Also the Bell transform of A001710(n+2) (adding 1,0,0,.. as column 0). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

			Triangle begins:
  [1],
  [3, 1],
  [12, 9, 1],
  [60, 75, 18, 1],
  [360, 660, 255, 30, 1],
  [2520, 6300, 3465, 645, 45, 1],
  ...

Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Wolfdieter Lang, First ten rows.
E. Neuwirth, Recursively defined combinatorial Functions: Extending Galton's board, Discr. Maths. 239 (2001) 33-51.
John Riordan, Letter, Apr 28 1976.

Cf. A001710, A049376, A105752.

Alternating row sums A134138.

a[n_, m_] /; n >= m >= 1 := a[n, m] = (2m + n - 1)*a[n-1, m] + a[n-1, m-1]; a[n_, m_] /; n < m = 0; a[, 0] = 0; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]] (* _Jean-François Alcover, Jul 22 2011 *)
a[n_, k_] := -(-1/2)^k*(n+1)!*HypergeometricPFQ[{1-k, n/2+1, (n+3)/2}, {3/2, 2}, 1]/(k-1)!; Table[a[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 28 2013, after Vladimir Kruchinin *)
a[0] = 0; a[n_] := (n + 1)!/2;
T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, a[0]^n], Sum[Binomial[n - 1, j - 1] a[j] T[n - j, k - 1], {j, 0, n - k + 1}]];
Table[T[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 19 2016, after Peter Luschny, updated Jan 01 2021 *)
rows = 9;
a[n_, m_] := BellY[n, m, Table[(k+2)!/2, {k, 0, rows}]];
Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

a(n,k):=(n!*sum((-1)^(k-j)*binomial(k,j)*binomial(n+2*j-1,2*j-1),j,1,k))/(2^k*k!); /* Vladimir Kruchinin, Apr 01 2011 */

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: factorial(n+2)//2, 9) # Peter Luschny, Jan 19 2016

a(n, m) = n!*A030523(n, m)/(m!*2^(n-m)); a(n, m) = (2*m+n-1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1; a(n, m)=0, n

a(n, m) = sum(|S1(n, j)|* A075497(j, m), j=m..n) (matrix product), with S1(n, j) := A008275(n, j) (signed Stirling1 triangle). Priv. comm. to Wolfdieter Lang by E. Neuwirth, Feb 15 2001; see also the 2001 Neuwirth reference.

a(n, k) = (n!*sum(j=1..k, (-1)^(k-j)*binomial(k,j)*binomial(n+2*j-1,2*j-1)))/(2^k*k!) - Vladimir Kruchinin, Apr 01 2011

New name from Peter Luschny, Jan 19 2016

A291709 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} (-1)^(j-1)binomial(-k,j-1)x^j/j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 6, 1, 1, 1, 4, 13, 24, 1, 1, 1, 5, 22, 73, 120, 1, 1, 1, 6, 33, 154, 501, 720, 1, 1, 1, 7, 46, 273, 1306, 4051, 5040, 1, 1, 1, 8, 61, 436, 2721, 12976, 37633, 40320, 1, 1, 1, 9, 78, 649, 4956, 31701, 147484, 394353, 362880, 1

Offset: 0

Seiichi Manyama, Oct 21 2017

			Square array B(j,k) begins:
   1,   1,   1,    1,    1, ...
   0,   1,   2,    3,    4, ...
   0,   1,   3,    6,   10, ...
   0,   1,   4,   10,   20, ...
   0,   1,   5,   15,   35, ...
   0,   1,   6,   21,   56, ...
Square array A(n,k) begins:
   1,   1,   1,    1,    1, ...
   1,   1,   1,    1,    1, ...
   1,   2,   3,    4,    5, ...
   1,   6,  13,   22,   33, ...
   1,  24,  73,  154,  273, ...
   1, 120, 501, 1306, 2721, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..10 give A000012, A000142, A000262, A049376, A049377, A049378, A049402, A132164, A293986, A293987, A293988.
Rows n=0-1 give A000012.
Main diagonal gives A293989.
Cf. A293012, A293991.

B[j_, k_] := (-1)^(j-1)*Binomial[-k, j-1];
A[0, ] = 1; A[n, k_] := (n-1)!*Sum[B[j, k]*A[n-j, k]/(n-j)!, {j, 1, n}];
Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 03 2017 *)

Let B(j,k) = (-1)^(j-1)*binomial(-k,j-1) for j>0 and k>=0.

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} B(j,k)*A(n-j,k)/(n-j)! for n > 0.

A157383 A partition product of Stirling_1 type [parameter k = -3] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 3, 1, 9, 12, 1, 45, 48, 60, 1, 165, 480, 300, 360, 1, 855, 3840, 3600, 2160, 2520, 1, 3843, 29400, 46200, 30240, 17640, 20160, 1, 21819, 272832, 520800, 443520, 282240, 161280, 181440

Offset: 1

Peter Luschny, Mar 07 2009, Mar 14 2009

Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = -3,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144353.
Same partition product with length statistic is A046089.
Diagonal a(A000217(n)) = rising_factorial(3,n-1), A001710(n+1).
Row sum is A049376.

Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_1 Triangles.

Cf. A157386, A157385, A157384, A157400, A126074, A157391, A157392, A157393, A157394, A157395

T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n
T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that
1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-2}(j-n-1).

A136658 Row sums of unsigned triangle A136656 and also of triangle 2*A136657.

Original entry on oeis.org

1, 2, 10, 68, 580, 5912, 69784, 933200, 13912336, 228390560, 4088594464, 79186453568, 1648396356160, 36678170613632, 868239454798720, 21776352497954048, 576629116655862016, 16069766602389885440, 470015788927133039104, 14392014594072635786240

Offset: 0

Wolfdieter Lang, Feb 22 2008

Alois P. Heinz, Table of n, a(n) for n = 0..432

Cf. A049376, A136656, A136657.

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*(j+1)!*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 01 2017

a[n_] := Sum[ StirlingS1[n, k] * BellB[k] * (-1)^(n-k) * 2^k, {k, 0, n}]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jul 09 2013, after Paul D. Hanna *)
Table[Sum[BellY[n, k, (Range[n] + 1)!], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}
{a(n)=sum(k=0, n, Stirling1(n, k)*Bell(k) * (-1)^(n-k)*2^k)}
/* Paul D. Hanna, Dec 25 2011 */

a(n) = Sum_{k=0..n} (-1)^n*A136656(n,k), n>=0.
E.g.f.: exp(x*(2-x)/(1-x)^2) (from Jabotinsky type triangle).
a(n) = Sum_{k=0..n} Stirling1(n, k) * Bell(k) * (-1)^(n-k) * 2^k. - Paul D. Hanna, Dec 25 2011
a(n) = (3*n-1)*a(n-1) - 3*(n-2)*(n-1)*a(n-2) + (n-3)*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Sep 25 2013
a(n) ~ 2^(1/6)*n^(n-1/6) * exp((n/2)^(1/3)+3*(n/2)^(2/3)-n-2/3) / sqrt(3) * (1 + 7/(27*(n/2)^(1/3)) - 422/(3645*(n/2)^(2/3))). - Vaclav Kotesovec, Sep 25 2013
Representation as special values of hypergeometric functions 2F2, in Maple notation: a(n) = (n+1)!*hypergeom([(1/2)*n+1, (1/2)*n+3/2], [3/2, 2], 1)*exp(-1), n = 1,2,... . - Karol A. Penson, Jul 28 2018

A144353 Partition number array, called M31(3), related to A046089(n,m)= |S1(3;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 3, 1, 12, 9, 1, 60, 48, 27, 18, 1, 360, 300, 360, 120, 135, 30, 1, 2520, 2160, 2700, 1440, 900, 2160, 405, 240, 405, 45, 1, 20160, 17640, 22680, 25200, 7560, 18900, 10080, 11340, 2100, 7560, 2835, 420, 945, 63, 1, 181440, 161280, 211680, 241920, 126000, 70560, 181440

Offset: 1

Wolfdieter Lang Oct 09 2008, Oct 28 2008

Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M31(3;n,k) with the k-th partition of n in A-St order.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
Third member (K=3) in the family M31(K) of partition number arrays.
If M31(3;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle |S1(3)|:= A046089.

			[1];[3,1];[12,9,1];[60,48,27,18,1];[360,300,360,120,135,30,1];...
a(4,3)= 27 = 3*|S1(3;2,1)|^2. The relevant partition of 4 is (2^2).

W. Lang, First 10 rows of the array and more.
W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

A049376 (row sums).

A130561 (M31(2) array), A144354 (M31(4) array).

a(n,k)=(n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S1(3;j,1)|^e(n,k,j),j=1..n) = M3(n,k)*product(|S1(3;j,1)|^e(n,k,j),j=1..n) with |S1(3;n,1)|= A001710(n+1) = (n+1)!/2!, n>=1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. M3(n,k)=A036040.

A380257 Expansion of e.g.f. exp( (1/(1-3*x)^(2/3) - 1)/2 ).

Original entry on oeis.org

1, 1, 6, 56, 706, 11186, 213156, 4742256, 120571676, 3447128796, 109427729096, 3818008773536, 145196289453656, 5976489668054296, 264685744187399536, 12548508890339297856, 634022724191046592016, 34007862777419093053456, 1929842567333195106456416

Offset: 0

Seiichi Manyama, Jan 18 2025

nonn

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A049376, A375173, A380258.

Cf. A004211, A380214.

CoefficientList[Series[Exp[ (1/(1-3*x)^(2/3) - 1)/2 ],{x,0,18}],x]Range[0,18]! (* Stefano Spezia, Mar 31 2025 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1/(1-3*x)^(2/3)-1)/2)))

a(n) = Sum_{k=0..n} 3^(n-k) * |Stirling1(n,k)| * A004211(k) = Sum_{k=0..n} 2^k * 3^(n-k) * |Stirling1(n,k)| * Bell_k(1/2), where Bell_n(x) is n-th Bell polynomial.

a(n) = (1/exp(1/2)) * (-3)^n * n! * Sum_{k>=0} binomial(-2*k/3,n)/(2^k * k!).

A380258 Expansion of e.g.f. exp( (1/(1-5*x)^(2/5) - 1)/2 ).

Original entry on oeis.org

1, 1, 8, 106, 1954, 46082, 1323064, 44750644, 1741897340, 76672512316, 3764746706176, 203976645319448, 12086590557877144, 777464693554778776, 53948773488864143072, 4016672567726156437744, 319379204127841984947472, 27010128651142535536409360, 2420802590890201251989984128

Offset: 0

Seiichi Manyama, Jan 18 2025

nonn

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A049376, A375173, A380257.

Cf. A004211, A025168.

CoefficientList[Series[Exp[ (1/(1-5*x)^(2/5) - 1)/2 ],{x,0,18}],x]Range[0,18]! (* Stefano Spezia, Mar 31 2025 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1/(1-5*x)^(2/5)-1)/2)))

a(n) = Sum_{k=0..n} 5^(n-k) * |Stirling1(n,k)| * A004211(k) = Sum_{k=0..n} 2^k * 5^(n-k) * |Stirling1(n,k)| * Bell_k(1/2), where Bell_n(x) is n-th Bell polynomial.

a(n) = (1/exp(1/2)) * (-5)^n * n! * Sum_{k>=0} binomial(-2*k/5,n)/(2^k * k!).

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A046089 Triangle read by rows, the Bell transform of (n+2)!/2 without column 0.

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A291709 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} (-1)^(j-1)*binomial(-k,j-1)*x^j/j).

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A157383 A partition product of Stirling_1 type [parameter k = -3] with biggest-part statistic (triangle read by rows).

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A136658 Row sums of unsigned triangle A136656 and also of triangle 2*A136657.

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A144353 Partition number array, called M31(3), related to A046089(n,m)= |S1(3;n,m)| (generalized Stirling triangle).

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A380257 Expansion of e.g.f. exp( (1/(1-3*x)^(2/3) - 1)/2 ).

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Programs

Mathematica

PARI

Formula

A380258 Expansion of e.g.f. exp( (1/(1-5*x)^(2/5) - 1)/2 ).

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Mathematica

PARI

Formula

A291709 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} (-1)^(j-1)binomial(-k,j-1)x^j/j).