A014322 -id:A014322 - cp's OEIS frontend

A342111 a(n) = (-1)^n * Sum_{k=0..n} Stirling1(n,k) * Stirling1(n,n-k).

1, 0, 1, 12, 193, 3980, 100805, 3034920, 105994833, 4215106728, 188097696345, 9309515255700, 506149663220641, 29989851619249236, 1923467938147053389, 132771455705186298000, 9814431285244231295265, 773520674985391641371280, 64752473306596841023424945

Offset: 0

Vaclav Kotesovec, Feb 28 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 0..350

Cf. A008275, A014322, A047796, A132393, A342110.
Cf. A048994, A155826.
Cf. A129256, A187235, A246117.

[(&+[(-1)^n*StirlingFirst(n, k)*StirlingFirst(n, n-k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jun 03 2021

Table[(-1)^n*Sum[StirlingS1[n, k]*StirlingS1[n, n-k], {k, 0, n}], {n, 0, 20}]

a(n) = (-1)^n*sum(k=0, n, stirling(n, k, 1)*stirling(n, n-k, 1)); \\ Michel Marcus, Feb 28 2021

[sum( stirling_number1(n, k)*stirling_number1(n, n-k) for k in (0..n) ) for n in (0..30)] # G. C. Greubel, Jun 03 2021

a(n) ~ c * d^n * (n-1)!, where
d = A238261 = -(2*LambertW(-1,-exp(-1/2)/2))^2 / (1 + 2*LambertW(-1,-exp(-1/2)/2)) = 4.9108149645682558987515348052403521978987052817678471761394112...
c = 1/(4*sqrt(-LambertW(-1, -exp(-1/2)/2)) * sqrt(-1 - LambertW(-1, -exp(-1/2)/2))*Pi) = 0.06903826111269387517867145566264007373042059749428879149076344304196548... - Vaclav Kotesovec, Feb 28 2021, updated May 14 2025
a(n) = [x^n] Product_{k=0..n-1} (1 + k*x)^2. - Seiichi Manyama, May 13 2025

A292870 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of k-th power of continued fraction 1/(1 - x - x^2/(1 - 2x - 2x^2/(1 - 3x - 3x^2/(1 - 4x - 4x^2/(1 - ...))))).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 5, 0, 1, 4, 9, 14, 15, 0, 1, 5, 14, 28, 44, 52, 0, 1, 6, 20, 48, 93, 154, 203, 0, 1, 7, 27, 75, 169, 333, 595, 877, 0, 1, 8, 35, 110, 280, 624, 1289, 2518, 4140, 0, 1, 9, 44, 154, 435, 1071, 2442, 5394, 11591, 21147, 0, 1, 10, 54, 208, 644, 1728, 4265, 10188, 24366, 57672, 115975, 0

Offset: 0

Ilya Gutkovskiy, Sep 25 2017

A(n,k) is the n-th term of the k-fold convolution of Bell numbers with themselves. - Alois P. Heinz, Feb 12 2019

			G.f. of column k: A_k(x) = 1 + k*x + k*(k + 3)*x^2/2 + k*(k^2 + 9*k + 20)*x^3/6 + k*(k^3 + 18*k^2 + 107*k + 234)*x^4/24 + k*(k^4 + 30*k^3 + 335*k^2 + 1770*k + 4104)*x^5/120 + ...
Square array begins:
  1,   1,    1,    1,    1,     1,  ...
  0,   1,    2,    3,    4,     5,  ...
  0,   2,    5,    9,   14,    20,  ...
  0,   5,   14,   28,   48,    75,  ...
  0,  15,   44,   93,  169,   280,  ...
  0,  52,  154,  333,  624,  1071,  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-4 give A000007, A000110, A014322, A014323, A014325.
Rows n=0-3 give A000012, A001477, A000096, A005586.
Antidiagonal sums give A137551.
Main diagonal gives A292871.
Cf. A205574 (another version).

A:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=0, 0,
     `if`(k=1, add(A(n-j, k)*binomial(n-1, j-1), j=1..n),
     (h-> add(A(j, h)*A(n-j, k-h), j=0..n))(iquo(k,2)))))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, May 31 2018

Table[Function[k, SeriesCoefficient[1/(1 - x + ContinuedFractionK[-i x^2, 1 - (i + 1) x, {i, 1, n}])^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

G.f. of column k: (1/(1 - x - x^2/(1 - 2*x - 2*x^2/(1 - 3*x - 3*x^2/(1 - 4*x - 4*x^2/(1 - ...))))))^k, a continued fraction.

A014323 Three-fold convolution of Bell numbers with themselves.

Original entry on oeis.org

1, 3, 9, 28, 93, 333, 1289, 5394, 24366, 118526, 618924, 3456942, 20573391, 129951231, 867877107, 6106194478, 45109290477, 348836705235, 2816093142803, 23673989688810, 206794355179656, 1873232870155036, 17565534522745008, 170237112831874188

Offset: 0

N. J. A. Sloane

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..576

Cf. A000110, A014322, A014325.

Column k=3 of A292870.

A014322:= func< n | (&+[Bell(j)*Bell(n-j): j in [0..n]]) >;
A014323:= func< n | (&+[Bell(j)*A014322(n-j): j in [0..n]]) >;
[A014323(n): n in [0..40]]; // G. C. Greubel, Jan 08 2023

A014322[n_]:= Sum[BellB[j]*BellB[n-j], {j,0,n}];
A014323[n_]:= Sum[BellB[j]*A014322[n-j], {j,0,n}];
Table[A014323[n], {n,0,40}] (* G. C. Greubel, Jan 08 2023 *)

def A014322(n): return sum(bell_number(j)*bell_number(n-j) for j in range(n+1))
def A014323(n): return sum(bell_number(j)*A014322(n-j) for j in range(n+1))
[A014323(n) for n in range(41)] # G. C. Greubel, Jan 08 2023

G.f.: (1/(1 - x - x^2/(1 - 2*x - 2*x^2/(1 - 3*x - 3*x^2/(1 - 4*x - 4*x^2/(1 - ...))))))^3, a continued fraction. - Ilya Gutkovskiy, Sep 25 2017
From G. C. Greubel, Jan 08 2023: (Start)
a(n) = Sum_{j=0..n} A000110(j)*A014322(n-j).
G.f.: ( Sum_{j>=0} A000110(j)*x^j )^3. (End)

A014325 Four-fold convolution of Bell numbers with themselves.

Original entry on oeis.org

1, 4, 14, 48, 169, 624, 2442, 10188, 45452, 217100, 1109914, 6064584, 35330715, 218788432, 1435302930, 9940062428, 72422364227, 553338786504, 4420324121772, 36820875272488, 319053830821880, 2869645346679368, 26739383194844404, 257682847299543248

Offset: 0

N. J. A. Sloane

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..575

Cf. A000110, A014322, A014323.

Column k=4 of A292870.

A014322:= func< n | (&+[Bell(j)*Bell(n-j): j in [0..n]]) >;
A014325:= func< n | (&+[A014322(j)*A014322(n-j): j in [0..n]]) >;
[A014325(n): n in [0..40]]; // G. C. Greubel, Jan 08 2023

A014322[n_]:= Sum[BellB[j]*BellB[n-j], {j,0,n}];
A014325[n_]:= Sum[A014322[j]*A014322[n-j], {j,0,n}];
Table[A014325[n], {n,0,40}] (* G. C. Greubel, Jan 08 2023 *)

def A014322(n): return sum(bell_number(j)*bell_number(n-j) for j in range(n+1))
def A014325(n): return sum(A014322(j)*A014322(n-j) for j in range(n+1))
[A014325(n) for n in range(41)] # G. C. Greubel, Jan 08 2023

G.f.: (1/(1 - x - x^2/(1 - 2*x - 2*x^2/(1 - 3*x - 3*x^2/(1 - 4*x - 4*x^2/(1 - ...))))))^4, a continued fraction. - Ilya Gutkovskiy, Sep 25 2017

G.f.: ( Sum_{j>=0} A000110(j)*x^j )^4. - G. C. Greubel, Jan 08 2023

A205574 Triangle T(n,k), 0<=k<=n, given by (0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 5, 3, 1, 0, 15, 14, 9, 4, 1, 0, 52, 44, 28, 14, 5, 1, 0, 203, 154, 93, 48, 20, 6, 1, 0, 877, 595, 333, 169, 75, 27, 7, 1, 0, 4140, 2518, 1289, 624, 280, 110, 35, 8, 1, 0, 21147, 11591, 5394, 2442, 1071, 435, 154, 44, 9, 1

Offset: 0

Philippe Deléham, Jan 29 2012

Bell convolution triangle ; g.f. for column k : (x*B(x))^k with  B(x) g.f. for A000110 (Bell numbers).
Riordan array (1, x*B(x)), when B(x) the g.f. of A000110.
Row sums are in A137551.

			Triangle begins:
  1;
  0,   1;
  0,   1,   1;
  0,   2,   2,  1;
  0,   5,   5,  3,  1;
  0,  15,  14,  9,  4,  1;
  0,  52,  44, 28, 14,  5, 1;
  0, 203, 154, 93, 48, 20, 6, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. Columns : A000007, A000110, A014322, A014323, A014325 ; Diagonals : A000012, A001477, A000096, A005586.
Another version: A292870.
T(2n,n) gives: A292871.

# Uses function PMatrix from A357368.
PMatrix(10, n -> combinat:-bell(n-1)); # Peter Luschny, Oct 19 2022

Sum_{k=0..n} T(n,k) = A137551(n), n>0.

A342110 a(n) = Sum_{k=0..n} Stirling2(n,k) * Stirling2(n,n-k).

Original entry on oeis.org

1, 0, 1, 6, 61, 770, 12160, 228382, 4989621, 124262532, 3475892685, 107901412520, 3681266754660, 136918473752216, 5513911474915116, 239034083286873630, 11098790133822288645, 549539910028075555016, 28903562131933534643851, 1609321474965547356327246

Offset: 0

Vaclav Kotesovec, Feb 28 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 0..350

Cf. A008277, A014322, A047797, A342111.

Cf. A048993.

[(&+[StirlingSecond(n, k)*StirlingSecond(n, n-k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jun 03 2021

Table[Sum[StirlingS2[n, k]*StirlingS2[n, n-k], {k, 0, n}], {n, 0, 20}]

a(n) = sum(k=0, n, stirling(n, k, 2)*stirling(n, n-k, 2)); \\ Michel Marcus, Feb 28 2021

[sum( stirling_number2(n, k)*stirling_number2(n, n-k) for k in (0..n) ) for n in (0..30)] # G. C. Greubel, Jun 03 2021

From Vaclav Kotesovec, Feb 28 2021, updated May 25 2025: (Start)
a(n) ~ c * d^n * (n-1)!, where
d = A238258 = -2 / (LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 3.0882773047417401791158400820254382768364448971420138767247...
c = 1/(2*Pi*sqrt((1 + LambertW(-2*exp(-2)))*(3 + LambertW(-2*exp(-2))))) = 0.12826577250734152801558828593238744179869387423941684693208180123477... (End)

A144155 Bell convolution triangle, T(n,k) = A000110(n-k)*A000110(k).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 5, 2, 2, 5, 15, 5, 4, 5, 15, 52, 15, 10, 10, 15, 52, 203, 52, 30, 25, 30, 52, 203, 877, 203, 104, 75, 75, 104, 203, 877, 4140, 877, 406, 260, 225, 260, 406, 877, 4140, 21147, 4140, 1754, 1015, 780, 780, 1015, 1754, 4140, 21147

Offset: 0

Gary W. Adamson, Sep 12 2008

Row sums = A014322: (1, 2, 5, 14, 44, 154,...) the Bell numbers convolved with themselves.

			First few rows of the triangle =
1;
1, 1;
2, 1, 2;
5, 2, 2, 5;
15, 5, 4, 5, 15;
52, 15, 10, 10, 15, 52;
203, 52, 30, 25, 30, 52, 203;
...
Row 3 = (5, 2, 2, 5) = termwise products of (1, 1, 2, 5) and (5, 2, 1, 1) = (5*1, 1*2, 2*1, 5*1).

A000110, Cf. A014322

Triangle read by rows, T(n,k) = A000110(n-k)*A000110(k)

More terms from Philippe Deléham, Jan 29 2012

cp's OEIS Frontend

A342111 a(n) = (-1)^n * Sum_{k=0..n} Stirling1(n,k) * Stirling1(n,n-k).

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A292870 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of k-th power of continued fraction 1/(1 - x - x^2/(1 - 2*x - 2*x^2/(1 - 3*x - 3*x^2/(1 - 4*x - 4*x^2/(1 - ...))))).

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A014323 Three-fold convolution of Bell numbers with themselves.

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A014325 Four-fold convolution of Bell numbers with themselves.

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A205574 Triangle T(n,k), 0<=k<=n, given by (0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A342110 a(n) = Sum_{k=0..n} Stirling2(n,k) * Stirling2(n,n-k).

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A144155 Bell convolution triangle, T(n,k) = A000110(n-k)*A000110(k).

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A292870 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of k-th power of continued fraction 1/(1 - x - x^2/(1 - 2x - 2x^2/(1 - 3x - 3x^2/(1 - 4x - 4x^2/(1 - ...))))).