A049425 -id:A049425 - cp's OEIS frontend

A049404 Triangle read by rows, the Bell transform of n!*binomial(2,n) (without column 0).

1, 2, 1, 2, 6, 1, 0, 20, 12, 1, 0, 40, 80, 20, 1, 0, 40, 360, 220, 30, 1, 0, 0, 1120, 1680, 490, 42, 1, 0, 0, 2240, 9520, 5600, 952, 56, 1, 0, 0, 2240, 40320, 48720, 15120, 1680, 72, 1, 0, 0, 0, 123200, 332640, 184800, 35280, 2760, 90, 1, 0, 0, 0, 246400, 1786400

Offset: 1

Wolfdieter Lang

Previous name was: A triangle of numbers related to triangle A049324.
a(n,1) = A008279(2,n-1). a(n,m) =: S1(-2; n,m), a member of a sequence of lower triangular Jabotinsky matrices, including S1(1; n,m) = A008275 (signed Stirling first kind), S1(2; n,m) = A008297(n,m) (signed Lah numbers).
a(n,m) matrix is inverse to signed matrix ((-1)^(n-m))*A004747(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).
For the definition of the Bell transform see A264428 and the link. - Peter Luschny, Jan 16 2016

			E.g. row polynomial E(3,x) = 2*x+6*x^2+x^3.
Triangle starts:
{1}
{2,  1}
{2,  6,  1}
{0, 20, 12, 1}

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
W. Lang, First 10 rows of the array and more. [From _Wolfdieter Lang_, Oct 17 2008]
Peter Luschny, The Bell transform

Row sums give A049425.

Cf. A004747, A049324.

rows = 11;
a[n_, m_] := BellY[n, m, Table[k! Binomial[2, k], {k, 0, rows}]];
Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

# uses[bell_matrix from A264428]
# Adds 1,0,0,0, ... as column 0 at the left side of the triangle.
bell_matrix(lambda n: factorial(n)*binomial(2, n), 8) # Peter Luschny, Jan 16 2016

a(n, m) = n!*A049324(n, m)/(m!*3^(n-m));
a(n, m) = (3*m-n+1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1;
a(n, m) = 0, nE.g.f. for m-th column: ((x+x^2+(x^3)/3)^m)/m!.
a(n,m) = n!/(3^m * m!)*(Sum_{i=0..floor(m-n/3)} (-1)^i * binomial(m,i) * binomial(3*m-3*i,n)), 0 for empty sums. - Werner Schulte, Feb 20 2020

New name from Peter Luschny, Jan 16 2016

A049377 Row sums of triangle A049352.

Original entry on oeis.org

1, 1, 5, 33, 273, 2721, 31701, 421905, 6302913, 104270913, 1889862021, 37204038081, 789866524305, 17977594555233, 436435929785493, 11251798888929201, 306889765901872641, 8825681949708120705, 266828094135981378693, 8458295877281844310113

Offset: 0

Wolfdieter Lang

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..428
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Column k=4 of A291709.

Cf. A049352, A049425.

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*(j+2)!/6*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 01 2017

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n-1, j-1]*(j+2)!/6*a[n-j], {j, 1, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

E.g.f. exp(p(x)) with p(x) := x*(3-3*x+x^2)/(3*(1-x)^3) (E.g.f. first column of A049352).
a(n) ~ n^(n-1/8)/2 * exp(-1/4 + 5*n^(1/4)/24 + sqrt(n)/2 + 4*n^(3/4)/3 - n). - Vaclav Kotesovec, Oct 23 2017
E.g.f.: Sum_{n>=0} ( Integral 1/(1-x)^4 dx )^n / n!, where the constant of integration is taken to be zero. - Paul D. Hanna, Apr 27 2019
From Seiichi Manyama, Jan 18 2025: (Start)
a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A004212(k).
a(n) = (1/exp(1/3)) * (-1)^n * n! * Sum_{k>=0} binomial(-3*k,n)/(3^k * k!). (End)

a(0)=1 prepended by Alois P. Heinz, Aug 01 2017

A144358 Partition number array, called M31(-2), related to A049404(n,m) = S1(-2;n,m) (generalized Stirling triangle).

Original entry on oeis.org

1, 2, 1, 2, 6, 1, 0, 8, 12, 12, 1, 0, 0, 40, 20, 60, 20, 1, 0, 0, 0, 40, 0, 240, 120, 40, 180, 30, 1, 0, 0, 0, 0, 0, 0, 280, 840, 0, 840, 840, 70, 420, 42, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2240, 0, 0, 1120, 6720, 1680, 0, 2240, 3360, 112, 840, 56, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2240, 0, 0

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(-2;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, ...].
First member (K=2) in the family M31(-K) of partition number arrays.
If M31(-2;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1(-2) := A049404.

			[1]; [2,1]; [2,6,1]; [0,8,12,12,1]; [0,0,40,20,60,20,1]; ...
a(4,3) = 12 = 3*S1(-2;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.

Cf. A049425 (row sums).

Cf. A144357 (M31(-1) array), A144877 (M31(-3) array).

a(n,k)=(n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(S1(-2;j,1)^e(n,k,j),j=1..n) = M3(n,k)*product(S1(-2;j,1)^e(n,k,j),j=1..n) with S1(-2;n,1)|= A008279(2,n-1)= [1,2,2,0,...], 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.

A249015 A binomial convolution.

Original entry on oeis.org

1, 1, 5, 17, 69, 339, 1677, 9321, 55137, 343659, 2285289, 15910857, 116120781, 886308147, 7033465989, 58008074409, 495792941337, 4381170220251, 39980186877537, 376025841184329, 3640077999981189, 36224841818288547, 370112212444620861, 3878334404076375657

Offset: 0

Emanuele Munarini, Oct 20 2014

nonn

Cf. A049425, A252284.

b[n_] := Sum[(n!/k!)Sum[Binomial[k,i]Binomial[k-i+2,n-2i-k]/3^i,{i,0,k}],{k,0,n}]
c[n_] := Sum[(n!/k!)(-1)^k Sum[Binomial[k,i]Binomial[k-i,n-2i-k]/3^i,{i,0,k}],{k,0,n}]
Table[If[n==0,1,0]+Sum[Binomial[n,k]b[k]c[n-k-1],{k,0,n-1}],{n,0,40}]

b(n) := sum((n!/k!)*sum(binomial(k,i)*binomial(k-i+2,n-2*i-k)/3^i,i,0,k),k,0,n);
c(n) := sum((n!/k!)*(-1)^k*sum(binomial(k,i)*binomial(k-i,n-2*i-k)/3^i,i,0,k),k,0,n);
makelist((if n=0 then 1 else 0)+sum(binomial(n,k)*b(k)*c(n-k-1),k,0,n-1),n,0,20);

a(n) = 0^0 + Sum_{k=0..n-1} binomial(n,k)*b(k)*c(n-k-1),
where the numbers b(n) = A049425(n+1) have e.g.f. (1+t)^2*exp(t+t^2+t^3/3)
and the numbers c(n) have e.g.f. exp(-(t+t^2+t^3/3)).
D-finite with recurrence: a(n+4) + n*a(n+3) - 3*(n+3)*a(n+2) - 3*(n+3)*(n+2)*a(n+1) - (n+3)*(n+2)*(n+1)*a(n) = 0.
E.g.f.: A(t) = 1+(1+t)^2*exp(t+t^2+t^3/3)*Integral_{u=0..t} exp(-(u+u^2+u^3/3)) du.
Differential equation for the e.g.f.: (1+t)*A''(t) - (2+3*t+3*t^2+t^3)*A'(t) - 3*(1+t)^2*A(t) = 0.

A369785 Expansion of e.g.f. exp( (exp(3*(exp(x)-1))-1)/3 ).

Original entry on oeis.org

1, 1, 5, 32, 252, 2368, 25865, 321310, 4461684, 68329293, 1142114917, 20663072796, 401891071075, 8355591197398, 184796601094141, 4329517995684305, 107060130166069859, 2785248872828731497, 76017344650249268158, 2171058618712177987046

Offset: 0

Seiichi Manyama, Feb 01 2024

nonn

Cf. A000258, A369784.

Cf. A004212, A049425.

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

a(n) = Sum_{k=0..n} Stirling2(n,k) * A004212(k).

A249014 A double binomial sum.

Original entry on oeis.org

1, 3, 15, 105, 933, 9951, 123123, 1727685, 27050985, 466795323, 8791179831, 179262508833, 3931730998605, 92237649141015, 2303515063987803, 60987344488950141, 1705641174191204433, 50228924171214633075, 1553143164997199612895

Offset: 0

Emanuele Munarini, Oct 20 2014

nonn

Cf. A049425.

Table[Sum[n!/k!Sum[Binomial[k,i]Binomial[n+k-i+1,2k+i+1]/3^i,{i,0,k}],{k,0,n}],{n,0,60}]

makelist(sum(n!/k!*sum(binomial(k,i)*binomial(n+k-i+1,2*k+i+1)/3^i,i,0,k),k,0,n),n,0,12);

E.g.f.: (1/(1-t)^2)*exp((3*t-3*t^2+t^3)/(3*(1-t)^3)).
a(n) = sum(n!/k!*sum(bin(k,i)*bin(n+k-i+1,2*k+i+1)/3^i,i=0..k),k=0..n).
a(n) = sum(Lah(n,k)*h(k),k=0..n), where Lah(n,k) are the Lah numbers and the numbers h(n) are defined by the e.g.f. h(x) = (1+t)^2*exp(t+t^2+t^3/3) (essentially sequence A049425).
a(n) = sum(Lah(n+1,k+1)*h(k),k=0..n), where Lah(n,k) are the Lah numbers and the numbers h(n) are defined by the e.g.f. h(x) = exp(t+t^2+t^3/3) (sequence A049425).
a(n) = sum(bin(n,k)*(n!/k!)*h(k),k=0..n), where the numbers h(n) are defined by the e.g.f. h(x) = (1+t)*exp(t+t^2+t^3/3).
Recurrence: a(n+4)-(4*n+15)*a(n+3)+6*(n+3)^2*a(n+2)-2*(n+3)*(n+2)*(2n+5)*a(n+1)+(n+3)*(n+2)^2*(n+1)*a(n)=0.

A249062 A double binomial sum.

Original entry on oeis.org

1, 2, 5, 18, 69, 306, 1497, 7890, 45033, 273474, 1760301, 11961522, 85265325, 636026418, 4947725889, 40019230386, 335868650577, 2918173355010, 26199114476373, 242657102748114, 2314964975130261, 22717352863875762, 229029972003647145, 2369438933865972498

Offset: 0

Emanuele Munarini, Oct 20 2014

nonn

Cf. A049425.

AList[n_] := CoefficientList[Series[(1 + t) E^(t + t^2 + t^3/3), {t, 0, n}], t] Table[k!, {k, 0, n}]
AList[100]

a(n) := sum((n!/k!)*sum(binomial(k,i)*binomial(k-i+1,n-k-2*i)/3^i,i,0,k),k,0,n);
makelist(a(n),n,0,24);

a(n) = sum((n!/k!)*sum(bin(k,i)*bin(k-i+1,n-k-2*i)/3^i,i=0..k),k=0..n).
E.g.f.: (1+t)*exp(t+t^2+t^3/3).
a(n+4)+(n+1)*a(n+3)-3*(n+3)*a(n+2)-3*(n+3)*(n+2)*a(n+1)-(n+3)*(n+2)*(n+1)*a(n)=0.

A380260 Expansion of e.g.f. exp( ((1+2*x)^(3/2) - 1)/3 ).

Original entry on oeis.org

1, 1, 2, 3, 9, 6, 111, -573, 7638, -95751, 1450431, -24643134, 468589617, -9843336567, 226448287794, -5662061186949, 152892006728841, -4434211761771978, 137468475061977663, -4536657554920874181, 158788359466681092966, -5875324355407515077439, 229142457698060305226367

Offset: 0

Seiichi Manyama, Jan 18 2025

sign

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A049425, A380259.

With[{nn=30},CoefficientList[Series[Exp[((1+2x)^(3/2)-1)/3],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 29 2025 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(((1+2*x)^(3/2)-1)/3)))

a(n) = Sum_{k=0..n} 2^(n-k) * Stirling1(n,k) * A004212(k) = Sum_{k=0..n} 3^k * 2^(n-k) * Stirling1(n,k) * Bell_k(1/3), where Bell_n(x) is n-th Bell polynomial.

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

Showing 1-10 of 10 results.

cp's OEIS Frontend

A157392 A partition product of Stirling_1 type [parameter k = 2] with biggest-part statistic (triangle read by rows).

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A293991 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..k+1} binomial(k,j-1)*x^j/j).

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A049404 Triangle read by rows, the Bell transform of n!*binomial(2,n) (without column 0).

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Mathematica

Sage

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A049377 Row sums of triangle A049352.

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Maple

Mathematica

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A144358 Partition number array, called M31(-2), related to A049404(n,m) = S1(-2;n,m) (generalized Stirling triangle).

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A249015 A binomial convolution.

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Mathematica

Maxima

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A369785 Expansion of e.g.f. exp( (exp(3*(exp(x)-1))-1)/3 ).

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PARI

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A249014 A double binomial sum.

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Mathematica

Maxima

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A249062 A double binomial sum.

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