A137341 -id:A137341 - cp's OEIS frontend

A350259 Triangle read by rows. T(n, k) = n! * BellPolynomial(n, k).

1, 0, 1, 0, 4, 12, 0, 30, 132, 342, 0, 360, 2256, 7416, 18144, 0, 6240, 54480, 223920, 651360, 1545600, 0, 146160, 1749600, 8892720, 30496320, 82911600, 193030560, 0, 4420080, 71638560, 446357520, 1792405440, 5552593200, 14460979680, 33232948560

Offset: 0

Peter Luschny, Dec 22 2021

			Triangle starts:
[0] 1
[1] 0,      1
[2] 0,      4,      12
[3] 0,     30,     132,     342
[4] 0,    360,    2256,    7416,    18144
[5] 0,   6240,   54480,  223920,   651360,  1545600
[6] 0, 146160, 1749600, 8892720, 30496320, 82911600, 193030560

Cf. A350256, A350257, A350258, A350260, A350261, A350262, A350263.

Cf. A000110, A137341.

A350259 := (n, k) -> ifelse(n = 0, 1, n! * BellB(n, k)):
seq(seq(A350259(n, k), k = 0..n), n = 0..7);

T[n_, k_] := n! BellB[n, k]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

A323128 Number T(n,k) of colored set partitions of [n] where elements of subsets have distinct colors and exactly k colors are used; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 4, 0, 1, 18, 30, 0, 1, 74, 360, 360, 0, 1, 310, 3450, 8880, 6240, 0, 1, 1382, 31770, 160080, 271800, 146160, 0, 1, 6510, 298662, 2635920, 8152200, 10190880, 4420080, 0, 1, 32398, 2918244, 42687960, 214527600, 468669600, 460474560, 166924800

Offset: 0

Alois P. Heinz, Aug 30 2019

			T(3,2) = 18: 1a|2a3b, 1a|2b3a, 1b|2a3b, 1b|2b3a, 1a3b|2a, 1b3a|2a, 1a3b|2b, 1b3a|2b, 1a2b|3a, 1b2a|3a, 1a2b|3b, 1b2a|3b, 1a|2a|3b, 1a|2b|3a, 1b|2a|3a, 1a|2b|3b, 1b|2a|3b, 1b|2b|3a.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,    4;
  0, 1,   18,     30;
  0, 1,   74,    360,     360;
  0, 1,  310,   3450,    8880,    6240;
  0, 1, 1382,  31770,  160080,  271800,   146160;
  0, 1, 6510, 298662, 2635920, 8152200, 10190880, 4420080;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Partition of a set

Columns k=0-1 give: A000007, A057427.
Row sums give A104600.
Main diagonal gives A137341.
T(2n,n) gives A324523.

A:= proc(n, k) option remember; `if`(n=0, 1, add(k!/(k-j)!
      *binomial(n-1, j-1)*A(n-j, k), j=1..min(k, n)))
    end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

A[n_, k_] := A[n, k] = If[n==0, 1, Sum[k!/(k - j)! Binomial[n - 1, j - 1]* A[n - j, k], {j, Min[k, n]}]];
T[n_, k_] := Sum[A[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 30 2020, after Alois P. Heinz *)

A152474 Triangle T(n,k) read by rows: Sum_{k=0..binomial(n,2)} T(n,k)q^k = n!Sum_{pi} faq(n,q)/Product_{i=1..n} e(i)!faq(i,q)^e(i), where pi runs over all nonnegative integer solutions to e(1) + 2e(2) + ... + n*e(n) = n and faq(i,q) = Product_{j=1..i} (q^j-1)/(q-1), i = 1..n.

Original entry on oeis.org

1, 1, 3, 1, 13, 8, 8, 1, 73, 63, 89, 78, 41, 15, 1, 501, 544, 909, 1095, 1200, 842, 680, 315, 129, 24, 1, 4051, 5225, 9734, 13799, 18709, 20441, 20520, 18101, 14831, 10200, 5891, 3199, 1109, 314, 35, 1, 37633, 55656, 112370, 177457, 270746, 352969, 442897

Offset: 0

Vladeta Jovovic, Dec 05 2008

			Triangle T(n,k) begins:
    1;
    1;
    3,   1;
   13,   8,   8,    1;
   73,  63,  89,   78,   41,  15,   1;
  501, 544, 909, 1095, 1200, 842, 680, 315, 129, 24, 1;
  ...

Alois P. Heinz, Rows n = 0..36, flattened

Cf. A000262 (first column), A105219(second column), A137341 (row sums), A152534.

T(n,n) gives A346981.

{T(n,k)=local(e_q=sum(j=0,n,x^j/prod(i=1,j,(q^i-1)/(q-1)))+x*O(x^n)); n!*polcoeff(polcoeff(exp(e_q-1),n,x)*prod(j=1,n,(q^j-1)/(q-1)),k,q)} \\ Paul D. Hanna, Dec 15 2008

Sum_{k=0..binomial(n,2)} T(n,k)*exp(2*Pi*I*k/n) = n!. - Vladeta Jovovic, Dec 05 2008
From Paul D. Hanna, Dec 15 2008: (Start)
E.g.f.: A(x,q) = exp(e_q(x,q) - 1) = Sum_{n>=0} Sum_{k=0..n(n-1)/2} T(n,k)*q^k*x^n/(n!*faq(n,q)) where e_q(x,q) = Sum_{n>=0} x^n/faq(n,q) and faq(n,q) = Product_{j=1..n} (q^j-1)/(q-1) with faq(0,q)=1.
Sum_{k=0..n(n-1)/2} T(n,k)*(-1)^k = n!*A000110((n+1)/2), where A000110 is the Bell numbers. (End)

T(0,0)=1 prepended by Alois P. Heinz, Feb 04 2018

A323099 Number T(n,k) of colored set partitions of [n] where exactly k colors are used for the elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 4, 0, 5, 30, 30, 0, 15, 210, 540, 360, 0, 52, 1560, 7800, 12480, 6240, 0, 203, 12586, 109620, 316680, 365400, 146160, 0, 877, 110502, 1583862, 7366800, 14733600, 13260240, 4420080, 0, 4140, 1051560, 23995440, 169011360, 521640000, 792892800, 584236800, 166924800

Offset: 0

Alois P. Heinz, Aug 30 2019

			Triangle T(n,k) begins:
  1;
  0,   1;
  0,   2,      4;
  0,   5,     30,      30;
  0,  15,    210,     540,     360;
  0,  52,   1560,    7800,   12480,     6240;
  0, 203,  12586,  109620,  316680,   365400,   146160;
  0, 877, 110502, 1583862, 7366800, 14733600, 13260240, 4420080;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Partition of a set

Columns k=0-1 give: A000007, A000110 (for n>0).
Row sums give A121017.
Main diagonal gives A137341.
Cf. A000142, A008277, A048993, A019538, A131689.

A:= proc(n, k) option remember; `if`(n=0, 1, add(
      A(n-j, k)*binomial(n-1, j-1)*k^j, j=1..n))
    end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);
# second Maple program:
T:= (n, k)-> combinat[bell](n)*Stirling2(n,k)*k!:
seq(seq(T(n, k), k=0..n), n=0..10);

A[n_, k_] := A[n, k] = If[n==0, 1, Sum[A[n-j, k] Binomial[n-1, j-1] k^j, {j, 1, n}]];
T[n_, k_] := Sum[A[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten
(* second program: *)
T[n_, k_] := BellB[n] StirlingS2[n, k] k!;
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 08 2020, after Alois P. Heinz *)

T(n,k) = Bell(n) * Sum_{i=0..k} (k-i)^n * (-1)^i * C(k,i).
T(n,k) = Bell(n) * A131689(n,k).
T(n,k) = Bell(n) * Stirling2(n,k) * k!.

A336140 Number of ways to choose a set partition of the parts of a strict integer composition of n.

Original entry on oeis.org

1, 1, 1, 5, 5, 9, 39, 43, 73, 107, 497, 531, 951, 1345, 2125, 8789, 9929, 16953, 24723, 38347, 52717, 219131, 240461, 419715, 600075, 938689, 1278409, 1928453, 6853853, 7815657, 13205247, 19051291, 29325121, 40353995, 60084905, 80722899, 277280079, 312239953

Offset: 0

Gus Wiseman, Jul 16 2020

nonn

A strict composition of n is a finite sequence of distinct positive integers summing to n.

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Set partitions are A000110.
Strict compositions are A032020.
Set partitions of binary indices are A050315.
Set partitions of strict partitions are A294617.
Cf. A000009, A001055, A008289, A035470, A063834, A072574, A137341, A279375.

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n$2, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 30 2020

Table[Sum[BellB[Length[ctn]],{ctn,Join@@Permutations/@Select[ IntegerPartitions[n],UnsameQ@@#&]}],{n,0,10}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0, If[n == 0,
     BellB[p]*p!, b[n, i-1, p] + b[n-i, Min[n-i, i-1], p+1]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 40] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

a(n) = Sum_{k = 0..n} A000110(k) * A072574(n,k) = Sum_{k = 0..n} k! * A000110(k) * A008289(n,k).

A274844 The inverse multinomial transform of A001818(n) = ((2*n-1)!!)^2.

Original entry on oeis.org

1, 8, 100, 1664, 34336, 843776, 24046912, 779780096, 28357004800, 1143189536768, 50612287301632, 2441525866790912, 127479926768287744, 7163315850315825152, 431046122080208896000, 27655699473265974050816, 1884658377677216933085184

Offset: 1

Johannes W. Meijer, Jul 27 2016

nonn

The inverse multinomial transform [IML] transforms an input sequence b(n) into the output sequence a(n). The IML transform inverses the effect of the multinomial transform [MNL], see A274760, and is related to the logarithmic transform, see A274805 and the first formula.
To preserve the identity MNL[IML[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the MNL.
In the a(n) formulas, see the examples, the cumulant expansion numbers A127671 appear.
We observe that the inverse multinomial transform leaves the value of a(0) undefined.
The Maple programs can be used to generate the inverse multinomial transform of a sequence. The first program is derived from a formula given by Alois P. Heinz for the logarithmic transform, see the first formula and A001187. The second program uses the e.g.f. for multivariate row polynomials, see A127671 and the examples. The third program uses information about the inverse of the inverse of the multinomial transform, see A274760.
The IML transform of A001818(n) = ((2*n-1)!!)^2 leads quite unexpectedly to A005411(n), a sequence related to certain Feynman diagrams.
Some IML transform pairs, n >= 1: A000110(n) and 1/A000142(n-1); A137341(n) and A205543(n); A001044(n) and A003319(n+1); A005442(n) and A000204(n); A005443(n) and A001350(n); A007559(n) and A000244(n-1); A186685(n+1) and A131040(n-1); A061711(n) and A141151(n); A000246(n) and A000035(n); A001861(n) and A141044(n-1)/A001710(n-1); A002866(n) and A000225(n); A000262(n) and A000027(n).

			Some a(n) formulas, see A127671:
a(0) = undefined
a(1) = (1/0!) * (1*x(1))
a(2) = (1/1!) * (1*x(2) - x(1)^2)
a(3) = (1/2!) * (1*x(3) - 3*x(2)*x(1) + 2*x(1)^3)
a(4) = (1/3!) * (1*x(4) - 4*x(3)*x(1) - 3*x(2)^2 + 12*x(2)*x(1)^2 - 6*x(1)^4)
a(5) = (1/4!) * (1* x(5) - 5*x(4)*x(1) - 10*x(3)*x(2) + 20*x(3)*x(1)^2 + 30*x(2)^2*x(1) -60*x(2)*x(1)^3 + 24*x(1)^5)

Richard P. Feynman, QED, The strange theory of light and matter, 1985.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric W. Weisstein MathWorld, Logarithmic Transform.
Wikipedia, Feynman diagram

Cf. A274760, A274805, A127671, A001187, A001818, A000079, A005411.

nmax:=17: b := proc(n): (doublefactorial(2*n-1))^2 end: c:= proc(n) option remember; b(n) - add(k*binomial(n, k)*b(n-k)*c(k), k=1..n-1)/n end: a := proc(n): c(n)/(n-1)! end: seq(a(n), n=1..nmax); # End first IML program.
nmax:=17: b := proc(n): (doublefactorial(2*n-1))^2 end: t1 := log(1+add(b(n)*x^n/n!, n=1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n*coeff(t2, x, n) end: seq(a(n), n=1..nmax); # End second IML program.
nmax:=17: b := proc(n): (doublefactorial(2*n-1))^2 end: f := series(exp(add(t(n)*x^n/n, n=1..nmax)), x, nmax+1): d := proc(n): n!*coeff(f, x, n) end: a(1):=b(1): t(1):= b(1): for n from 2 to nmax+1 do t(n) := solve(d(n)-b(n), t(n)): a(n):=t(n): od: seq(a(n), n=1..nmax); # End third IML program.

nMax = 22; b[n_] := ((2*n-1)!!)^2; c[n_] := c[n] = b[n] - Sum[k*Binomial[n, k]*b[n-k]*c[k], {k, 1, n-1}]/n; a[n_] := c[n]/(n-1)!; Table[a[n], {n, 1, nMax}] (* Jean-François Alcover, Feb 27 2017, translated from Maple *)

a(n) = c(n)/(n-1)! with c(n) = b(n) - Sum_{k=1..n-1}(k*binomial(n, k)*b(n-k)*c(k)), n >= 1 and a(0) = undefined, with b(n) = A001818(n) = ((2*n-1)!!)^2.

a(n) = A000079(n-1) * A005411(n), n >= 1.

A156788 Triangle T(n, k) = binomial(n, k)A000166(n-k)k^n with T(0, 0) = 1, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 4, 0, 3, 0, 27, 0, 8, 96, 0, 256, 0, 45, 640, 2430, 0, 3125, 0, 264, 8640, 29160, 61440, 0, 46656, 0, 1855, 118272, 688905, 1146880, 1640625, 0, 823543, 0, 14832, 1899520, 16166304, 41287680, 43750000, 47029248, 0, 16777216, 0, 133497, 34172928, 438143580, 1453326336, 2214843750, 1693052928, 1452729852, 0, 387420489

Offset: 0

Roger L. Bagula, Feb 15 2009

			Triangle begins as:
  1;
  0,     1;
  0,     0,       4;
  0,     3,       0,       27;
  0,     8,      96,        0,      256;
  0,    45,     640,     2430,        0,     3125;
  0,   264,    8640,    29160,    61440,        0,    46656;
  0,  1855,  118272,   688905,  1146880,  1640625,        0, 823543;
  0, 14832, 1899520, 16166304, 41287680, 43750000, 47029248,      0, 16777216;

J. Riordan, Combinatorial Identities, Wiley, 1968, p.194.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000166, A000240, A000312, A137341.

A000166[n_]:= A000166[n]= If[n==0, 1, n*A000166[n-1] + (-1)^n];
T[n_, k_]:= If[n==0, 1, Binomial[n, k]*A000166[n-k]*k^n];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jun 10 2021 *)

def A000166(n): return 1 if (n==0) else n*A000166(n-1) + (-1)^n
def A156788(n,k): return 1 if (n==0) else binomial(n,k)*k^n*A000166(n-k)
flatten([[A156788(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 10 2021

T(n, k) = binomial(n, k)*A000166(n-k)*k^n with T(0, 0) = 1.
T(n, k) = binomial(n, k)*b(n-k)*k^n, where b(n) = n*b(n-1) + (-1)^n and b(0) = 1.
Sum_{k=0..n} T(n, k) = A137341(n).
From G. C. Greubel, Jun 10 2021: (Start)
T(n, 1) = A000240(n).
T(n, n) = A000312(n). (End)

Edited by G. C. Greubel, Jun 10 2021

A156789 Irregular triangle, read by rows, T(n, k) = binomial(2n, k)binomial(2*k, k).

Original entry on oeis.org

1, 1, 4, 6, 1, 8, 36, 80, 70, 1, 12, 90, 400, 1050, 1512, 924, 1, 16, 168, 1120, 4900, 14112, 25872, 27456, 12870, 1, 20, 270, 2400, 14700, 63504, 194040, 411840, 579150, 486200, 184756, 1, 24, 396, 4400, 34650, 199584, 853776, 2718144, 6370650, 10696400, 12193896, 8465184, 2704156

Offset: 0

Roger L. Bagula, Feb 15 2009

Row sums are A137341: {1, 11, 195, 3989, 86515, 1936881, 44241261, 1024642875, 23973456915, 565280386625, 13411044301945, ...}.

			Triangle begins as:
  1;
  1,  4,   6;
  1,  8,  36,   80,    70;
  1, 12,  90,  400,  1050,  1512,    924;
  1, 16, 168, 1120,  4900, 14112,  25872,  27456,  12870;
  1, 20, 270, 2400, 14700, 63504, 194040, 411840, 579150, 486200, 184756;

J. Riordan, Combinatorial Identities, Wiley, 1968, p.77.

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Flat(List([0..10], n-> List([0..2*n], k->Binomial(2*n, k)*Binomial(2*k, k) ))); # G. C. Greubel, Nov 30 2019

[Binomial(2*n, k)*Binomial(2*k, k): k in [0..2*n], n in [0..10]]; // G. C. Greubel, Nov 30 2019

seq(seq( binomial(2*n, k)*binomial(2*k, k), k=0..2*n), n=0..10); # G. C. Greubel, Nov 30 2019

Table[Binomial[2*n, k]*Binomial[2*k, k], {n,0,10}, {k,0,2*n}]//Flatten

T(n,k) = binomial(2*n, k)*binomial(2*k, k); \\ G. C. Greubel, Nov 30 2019

[[binomial(2*n, k)*binomial(2*k, k) for k in (0..2*n)] for n in (0..10)] # G. C. Greubel, Nov 30 2019

T(n, k) = binomial(2*n, k)*binomial(2*k, k).

A307362 Expansion of e.g.f. Sum_{j>=0} (exp(x) - 1)^j / Product_{k=1..j} (1 - k*(exp(x) - 1)).

Original entry on oeis.org

1, 1, 5, 43, 569, 10651, 265985, 8498323, 336759449, 16158195691, 920970111665, 61390084384003, 4724023128773129, 415070770350493531, 41252331696522595745, 4599993183150111332083, 571422442346267636255609, 78581827113539181495412171, 11896744343184751608550862225

Offset: 0

Ilya Gutkovskiy, Apr 05 2019

nonn

Cf. A000110, A000258, A000670, A137341, A307363.

nmax = 18; CoefficientList[Series[Sum[(Exp[x] - 1)^j/Product[(1 - k (Exp[x] - 1)), {k, 1, j}], {j, 0, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 18; CoefficientList[Series[Sum[j! BellB[j] x^j/Product[(1 - k x), {k, 1, j}], {j, 0, nmax}], {x, 0, nmax}], x]
Table[Sum[StirlingS2[n, k] k! BellB[k], {k, 0, n}], {n, 0, 18}]

G.f.: Sum_{j>=0} j!*Bell(j)*x^j / Product_{k=1..j} (1 - k*x).

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

A307363 Expansion of e.g.f. Sum_{j>=0} log(1 + x)^j / Product_{k=1..j} (1 - k*log(1 + x)).

Original entry on oeis.org

1, 1, 3, 20, 218, 3514, 77386, 2220504, 80085792, 3533917704, 186779329704, 11623513158960, 839754709300800, 69603737430736560, 6552428441847854640, 694531396130434062720, 82265733994694038784640, 10816812417663289139328000, 1569560370536552329095091200

Offset: 0

Ilya Gutkovskiy, Apr 05 2019

nonn

Cf. A000110, A006252, A137341, A307362.

nmax = 18; CoefficientList[Series[Sum[Log[1 + x]^j/Product[(1 - k Log[1 + x]), {k, 1, j}], {j, 0, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k! BellB[k], {k, 0, n}], {n, 0, 18}]

a(n) = Sum_{k=0..n} Stirling1(n,k)*k!*Bell(k).

cp's OEIS Frontend

A350259 Triangle read by rows. T(n, k) = n! * BellPolynomial(n, k).

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A323128 Number T(n,k) of colored set partitions of [n] where elements of subsets have distinct colors and exactly k colors are used; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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Maple

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A152474 Triangle T(n,k) read by rows: Sum_{k=0..binomial(n,2)} T(n,k)*q^k = n!*Sum_{pi} faq(n,q)/Product_{i=1..n} e(i)!*faq(i,q)^e(i), where pi runs over all nonnegative integer solutions to e(1) + 2*e(2) + ... + n*e(n) = n and faq(i,q) = Product_{j=1..i} (q^j-1)/(q-1), i = 1..n.

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PARI

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A323099 Number T(n,k) of colored set partitions of [n] where exactly k colors are used for the elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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Maple

Mathematica

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A336140 Number of ways to choose a set partition of the parts of a strict integer composition of n.

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Maple

Mathematica

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A274844 The inverse multinomial transform of A001818(n) = ((2*n-1)!!)^2.

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Maple

Mathematica

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A156788 Triangle T(n, k) = binomial(n, k)*A000166(n-k)*k^n with T(0, 0) = 1, read by rows.

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Mathematica

Sage

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A156789 Irregular triangle, read by rows, T(n, k) = binomial(2*n, k)*binomial(2*k, k).

A152474 Triangle T(n,k) read by rows: Sum_{k=0..binomial(n,2)} T(n,k)q^k = n!Sum_{pi} faq(n,q)/Product_{i=1..n} e(i)!faq(i,q)^e(i), where pi runs over all nonnegative integer solutions to e(1) + 2e(2) + ... + n*e(n) = n and faq(i,q) = Product_{j=1..i} (q^j-1)/(q-1), i = 1..n.

A156788 Triangle T(n, k) = binomial(n, k)A000166(n-k)k^n with T(0, 0) = 1, read by rows.

A156789 Irregular triangle, read by rows, T(n, k) = binomial(2n, k)binomial(2*k, k).