A001715 -id:A001715 - cp's OEIS frontend

A094646 Generalized Stirling number triangle of first kind.

1, -2, 1, 2, -3, 1, 0, 2, -3, 1, 0, 2, -1, -2, 1, 0, 4, 0, -5, 0, 1, 0, 12, 4, -15, -5, 3, 1, 0, 48, 28, -56, -35, 7, 7, 1, 0, 240, 188, -252, -231, 0, 42, 12, 1, 0, 1440, 1368, -1324, -1638, -231, 252, 114, 18, 1, 0, 10080, 11016, -7900, -12790, -3255, 1533, 1050, 240, 25, 1

Offset: 0

Vladeta Jovovic, May 17 2004

Triangle T(n,k), 0 <= k <= n, read by rows, given by [ -2, 1, -1, 2, 0, 3, 1, 4, 2, 5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 23 2006
From Wolfdieter Lang, Jun 23 2011: (Start)
The row polynomials s(n,x):=Sum_{k=0..n} T(n,k)*x^k satisfy risefac(x-2,n)=s(n,x), with the rising factorials risefac(x-2,n):=Product_{j=0..n-1} (x-2+j), n >= 1, risefac(x-2,0)=1. Compare this with the formula risefac(x,n)=|S1|(n,x), with the row polynomials |S1|(n,x) of A132393 (unsigned Stirling1).
This is the third triangle of an a-family of Sheffer arrays, call them |S1|(a), with e.g.f. of the row polynomials |S1|(a;x;z) = ((1-z)^a)*exp(-x*log(1-z)). In the notation showing the column e.g.f.s this is Sheffer ((1-z)^a,-log(1-z)). In the umbral notation (see the Roman reference, given under A094645) this is called Sheffer for (exp(a*t),1-exp(-t)). For a=0 this becomes the unsigned Stirling1 triangle |S1|(0) = A132393 with row polynomials |S1|(0;n,x) =: s1(n,x).
E.g.f. column number k (with leading zeros): ((1-x)^a)*((-log(1-x))^k)/k!, k >= 0.
E.g.f. for row sums is (1-x)^(a-1), and the e.g.f. for the alternating row sums is (1-x)^(a+1).
Row polynomial recurrence:
  |S1|(a;n,x)=(x+(n-1-a))*|S1|(a;n-1,m), |S1|(a;0,x)=1.
Meixner identity (see the reference under A060338):
  |S1|(a;n,x) - |S1|(a;n,x-1) = n*|S1|(a;n-1,x), n >= 1,
Also (from the corollary 3.7.2 on p. 50 of the Roman reference): |S1|(a;n,x) = (x-a)*|S1|(a;n-1,x+1), n >= 1.
Recurrence: |S1|(a;n,k) = |S1|(a;n-1,k-1) + (n-(a+1))*|S1|(a;n-1,k); |S1|(a;n,k)=0 if n < m, |S1|(a;n,-1)=0, |S1|(a;0,0)=1.
Connection to |Stirling1|=|S1|(0):
  |S1|(a;n,k) = Sum_{p=0..a} |S1|(a;a,p)*abs(Stirling1(n-a,k-p)), n >= a.
The exponential convolution identity is
  |S1|(a;n,x+y) = Sum_{k=0..n} binomial(n,k)*|S1|(a;k,y)*s1(n-k,x), n >= 0, with symmetry x <-> y.
The Sheffer a- and z-sequences are (see the W. Lang link under A006232): Sha(a;n)=A164555(n)/A027642(n) (independent of a) with e.g.f. x/(1-exp(-x)), and the z-sequence has e.g.f. (exp(a*x)-1)/(exp(-x)-1).
The inverse Sheffer matrix has e.g.f. exp(a*z)*exp(x*(1-exp(-z))), in short notation (exp(a*z),1-exp(-z)),
  (or in umbral notation ((1-t)^a,-log(1-t))).
(End)

			Triangle begins
   1;
  -2,  1;
   2, -3,  1;
   0,  2, -3,  1;
   0,  2, -1, -2,  1;
   0,  4,  0, -5,  0,  1;
   ...
risefac(x-2,3) = (x-2)*(x-1)*x = 2*x-3*x^2+x^3.
-1 = T(4,2) = T(3,1) + 1*T(3,2) =  2 + (-3).
T(4,3) = 2*abs(S1(2,3)) - 3*abs(S1(2,2)) + 1*abs(S1(2,1)) = 2*0 - 3*1 + 1*1 = -2.

Cf. A049444, A049458, A094645.

A094646_row := n -> seq((-1)^(n-k)*coeff(expand(pochhammer(x-n+3, n)), x, k), k=0..n): seq(print(A094646_row(n)), n = 0..6); # Peter Luschny, May 16 2013

Flatten[ Table[ CoefficientList[ Pochhammer[x-2, n], x], {n, 0, 10}]] (* Jean-François Alcover, Sep 26 2011 *)

E.g.f.: (1-y)^(2-x).
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000142(n), A000142(n+1), A001710(n+2), A001715(n+3), A001720(n+4), A001725(n+5), A001730(n+6), A049388(n), A049389(n), A049398(n), A051431(n) for x = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 respectively. - Philippe Deléham, Nov 13 2007
If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-a-j), then |T(n,i)| = |f(n,i,-2)|, for n=1,2,...; i=0..n. - Milan Janjic, Dec 21 2008
From Wolfdieter Lang, Jun 23 2011: (Start)
risefac(x-2,n) = Sum_{k=0..n} T(n,k)*x^k, n >= 0, with the rising factorials (see a comment above).
Recurrence: T(n,k) = T(n-1,k-1) + (n-3)*T(n-1,k); T(n,k)=0 if n < m, T(n,-1)=0, T(0,0)=1.
T(n,k) = 2*abs(S1(n-2,k)) - 3*abs(S1(n-2,k-1)) + abs(S1(n-2,k-2)), n >= 2, with S1(n,k) = Stirling1(n,k) = A048994(n,k).
E.g.f. column number k (with leading zeros):
((1-x)^2)*((-log(1-x))^k)/k!, k >= 0.
E.g.f. for row sums is 1-x, i.e., [1,-1,0,0,...],
  and the e.g.f. for the alternating row sums is (1-x)^3. i.e., [1,-3,3,1,0,0,...]. (End)

A144354 Partition number array, called M31(4), related to A049352(n,m)= |S1(4;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 4, 1, 20, 12, 1, 120, 80, 48, 24, 1, 840, 600, 800, 200, 240, 40, 1, 6720, 5040, 7200, 4000, 1800, 4800, 960, 400, 720, 60, 1, 60480, 47040, 70560, 84000, 17640, 50400, 28000, 33600, 4200, 16800, 6720, 700, 1680, 84, 1, 604800, 483840, 752640, 940800, 504000, 188160

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(4;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,...].
Fourth member (K=4) in the family M31(K) of partition number arrays.
If M31(4;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle |S1(4)|:= A049352.

			[1];[4,1];[20,12,1];[120,80,48,24,1];[840,600,800,200,240,40,1];...
a(4,3)= 48 = 3*|S1(4;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.

A049377 (row sums).

A144353 (M31(3) array), A144355 (M31(5) array).

a(n,k)=(n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S1(4;j,1)|^e(n,k,j),j=1..n) = M3(n,k)*product(|S1(4;j,1)|^e(n,k,j),j=1..n) with |S1(4;n,1)|= A001715(n+2) = (n+2)!/3!, 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.

A144885 Partition number array, called M31hat(4).

Original entry on oeis.org

1, 4, 1, 20, 4, 1, 120, 20, 16, 4, 1, 840, 120, 80, 20, 16, 4, 1, 6720, 840, 480, 400, 120, 80, 64, 20, 16, 4, 1, 60480, 6720, 3360, 2400, 840, 480, 400, 320, 120, 80, 64, 20, 16, 4, 1, 604800, 60480, 26880, 16800, 14400, 6720, 3360, 2400, 1920, 1600, 840, 480, 400, 320

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) =: M31hat(4;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,...].
Fourth member (K=4) in the family M31hat(K) of partition number arrays.
If M31hat(4;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1hat(4):= A144886.

			[1];[4,1];[20,4,1];[120,20,16,4,1];[840,120,80,20,16,4,1];...
a(4,3)= 16 = |S1(4;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.

A144887 (row sums).

A144880 (M31hat(3) array). A144886 (S1hat(4)).

a(n,k) = product(|S1(4;j,1)|^e(n,k,j),j=1..n) with |S1(4;n,1)| = A049352(n,1) = A001715(n+2) = [1,4,20,120,840,6720,...] = (n+2)!/3!, 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.

A172455 The case S(6,-4,-1) of the family of self-convolutive recurrences studied by Martin and Kearney.

Original entry on oeis.org

1, 7, 84, 1463, 33936, 990542, 34938624, 1445713003, 68639375616, 3676366634402, 219208706540544, 14397191399702118, 1032543050697424896, 80280469685284582812, 6725557192852592984064, 603931579625379293509683

Offset: 1

N. J. A. Sloane, Nov 20 2010

nonn

			G.f. = x + 7*x^2 + 84*x^3 + 1463*x^4 + 33936*x^5 + 990542*x^6 + 34938624*x^7 + ...
a(2) = 7 since (6*2 - 4) * a(2-1) - (a(1) * a(2-1)) = 7.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 307.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, arXiv:1103.4936 [math.CO], 2011.
NIST Digital Library of Mathematical Functions, Airy Functions.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
Eric Weisstein's World of Mathematics, Airy Functions, contains the definitions of Ai(x), Bi(x).

Cf. A000079 S(1,1,-1), A000108 S(0,0,1), A000142 S(1,-1,0), A000244 S(2,1,-2), A000351 S(4,1,-4), A000400 S(5,1,-5), A000420 S(6,1,-6), A000698 S(2,-3,1), A001710 S(1,1,0), A001715 S(1,2,0), A001720 S(1,3,0), A001725 S(1,4,0), A001730 S(1,5,0), A003319 S(1,-2,1), A005411 S(2,-4,1), A005412 S(2,-2,1), A006012 S(-1,2,2), A006318 S(0,1,1), A047891 S(0,2,1), A049388 S(1,6,0), A051604 S(3,1,0), A051605 S(3,2,0), A051606 S(3,3,0), A051607 S(3,4,0), A051608 S(3,5,0), A051609 S(3,6,0), A051617 S(4,1,0), A051618 S(4,2,0), A051619 S(4,3,0), A051620 S(4,4,0), A051621 S(4,5,0), A051622 S(4,6,0), A051687 S(5,1,0), A051688 S(5,2,0), A051689 S(5,3,0), A051690 S(5,4,0), A051691 S(5,5,0), A053100 S(6,1,0), A053101 S(6,2,0), A053102 S(6,3,0), A053103 S(6,4,0), A053104 S(7,1,0), A053105 S(7,2,0), A053106 S(7,3,0), A062980 S(6,-8,1), A082298 S(0,3,1), A082301 S(0,4,1), A082302 S(0,5,1), A082305 S(0,6,1), A082366 S(0,7,1), A082367 S(0,8,1), A105523 S(0,-2,1), A107716 S(3,-4,1), A111529 S(1,-3,2), A111530 S(1,-4,3), A111531 S(1,-5,4), A111532 S(1,-6,5), A111533 S(1,-7,6), A111546 S(1,0,1), A111556 S(1,1,1), A143749 S(0,10,1), A146559 S(1,1,-2), A167872 S(2,-3,2), A172450 S(2,0,-1), A172485 S(-1,-2,3), A177354 S(1,2,1), A292186 S(4,-6,1), A292187 S(3, -5, 1).

a[1] = 1; a[n_]:= a[n] = (6*n-4)*a[n-1] - Sum[a[k]*a[n-k], {k, 1, n-1}]; Table[a[n], {n, 1, 20}] (* Vaclav Kotesovec, Jan 19 2015 *)

{a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (6 * k - 4) * A[k-1] - sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 24 2011 */

S(v1, v2, v3, N=16) = {
  my(a = vector(N)); a[1] = 1;
  for (n = 2, N, a[n] = (v1*n+v2)*a[n-1] + v3*sum(j=1,n-1,a[j]*a[n-j])); a;
};
S(6,-4,-1)
\\ test: y = x*Ser(S(6,-4,-1,201)); 6*x^2*y' == y^2 - (2*x-1)*y - x
\\ Gheorghe Coserea, May 12 2017

a(n) = (6*n - 4) * a(n-1) - Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 24 2011
G.f.: x / (1 - 7*x / (1 - 5*x / (1 - 13*x / (1 - 11*x / (1 - 19*x / (1 - 17*x / ... )))))). - Michael Somos, Jan 03 2013
a(n) = 3/(2*Pi^2)*int((4*x)^((3*n-1)/2)/(Ai'(x)^2+Bi'(x)^2), x=0..inf), where Ai'(x), Bi'(x) are the derivatives of the Airy functions. [Vladimir Reshetnikov, Sep 24 2013]
a(n) ~ 6^n * (n-1)! / (2*Pi) [Martin + Kearney, 2011, p.16]. - Vaclav Kotesovec, Jan 19 2015
6*x^2*y' = y^2 - (2*x-1)*y - x, where y(x) = Sum_{n>=1} a(n)*x^n. - Gheorghe Coserea, May 12 2017
G.f.: x/(1 - 2*x - 5*x/(1 - 7*x/(1 - 11*x/(1 - 13*x/(1 - ... - (6*n - 1)*x/(1 - (6*n + 1)*x/(1 - .... Cf. A062980. - Peter Bala, May 21 2017

A129923 a(n) = (n+5)! / 5.

Original entry on oeis.org

24, 144, 1008, 8064, 72576, 725760, 7983360, 95800320, 1245404160, 17435658240, 261534873600, 4184557977600, 71137485619200, 1280474741145600, 24329020081766400, 486580401635328000, 10218188434341888000

Offset: 0

Roger L. Bagula, Jun 06 2007

G. C. Greubel, Table of n, a(n) for n = 0..440

Cf. A001710, A001715, A002301, A133799.

[Factorial(n+5)/5: n in [0..30]]; // G. C. Greubel, Sep 28 2024

Table[(n+5)!/5, {n, 0, 5 + 25}]
Range[5,30]!/5 (* Harvey P. Dale, Mar 29 2023 *)

PARI
```
a(n)=(n+5)!/5;
```

[factorial(n+5)//5 for n in range(31)] # G. C. Greubel, Sep 28 2024

E.g.f.: 24/(1 - x)^6. - G. C. Greubel, Sep 28 2024

Edited by Ralf Stephan, Nov 10 2013

A144888 Second column (m=2) of triangle A144886 (S1hat(4)).

Original entry on oeis.org

1, 4, 36, 200, 1720, 12480, 118560, 1081920, 11793600, 131443200, 1658764800, 21990528000, 319711795200, 4922394624000, 81508654080000, 1428114530304000, 26582538673152000, 521466739605504000, 10779099461222400000, 233753593186713600000, 5310546788872765440000

Offset: 0

Wolfdieter Lang, Oct 09 2008

Cf. A144886, A001715 (first column), A144889 (third column).

a(n) = A144886(n+2,2), n>=0.

A158777 Irregular array T(n,k), read by rows: row n is the polynomial expansion in t of p(x,t) = exp(tx)/(1 - x/t - t^4 x^4) with weighting factors t^n*n!.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 2, 0, 1, 6, 0, 6, 0, 3, 0, 1, 24, 0, 24, 0, 12, 0, 4, 0, 25, 120, 0, 120, 0, 60, 0, 20, 0, 245, 0, 121, 720, 0, 720, 0, 360, 0, 120, 0, 2190, 0, 1446, 0, 361, 5040, 0, 5040, 0, 2520, 0, 840, 0, 20370, 0, 15162, 0, 5047, 0, 841, 40320, 0, 40320, 0, 20160, 0, 6720, 0

Offset: 0

Roger L. Bagula, Mar 26 2009

Row sums are A334157: {1, 2, 5, 16, 89, 686, 5917, 54860, 588401, 7370074, ...}.
Outer diagonal is A330045: {1, 1, 1, 1, 25, 121, 361, 841, 42001, 365905, ...}.
From Petros Hadjicostas, Apr 15 2020: (Start)
To prove the general formula below for T(n,2*m), let v = x/t in the equation Sum_{n,k >= 0} (T(n,k)/n!) * (x/t)^n * t^k = p(x,t). We get Sum_{n,k >= 0} (T(n,k)/n!) * v^n * t^k = exp(t^2*v)/(1 - v - t^8*v^4).
Using the Taylor expansions of exp(t^2*v) and 1/(1 - v - t^8*v^4) around v = 0 (from array A180184), we get that Sum_{n,k >= 0} (T(n,k)/n!) * v^n * t^k = (Sum_{n >= 0} (t^2*v)^n/n!) * (Sum_{n >= 0} Sum_{h=0..floor(n/4)} binomial(n - 3*h, h)*t^(8*h)*v^n).
Using the Cauchy product of the above two series, for each n >= 0, we get Sum_{k >= 0} (T(n,k)/n!)*t^k = Sum_{l=0..n} Sum_{h=0..floor(l/4)} (binomial(l - 3*h, h)/(n-l)!)*t^(8*h+2*n-2*l). This implies that T(n,k) = 0 for odd k >= 1.
Letting k = 2*m = 8*h + 2*n - 2*l and s = n - l, we get Sum_{m >= 0} (T(n, 2*m)/n!)*t^(2*m) = Sum_{m >= 0} Sum_{s=0..m with 4|(m-s)} (binomial(n - s - 3*(m - s)/4, (m - s)/4)/s!)*t^(2*m) (and also that T(n,2*j) = 0 for j > m). Equating coefficients, we get the formula for T(n,2*m) shown below. (End)

			Array T(n,k) (with n >= 0 and 0 <= k <= 2*n) begins as follows:
     1;
     1, 0,    1;
     2, 0,    2, 0,    1;
     6, 0,    6, 0,    3, 0,   1;
    24, 0,   24, 0,   12, 0,   4, 0,    25;
   120, 0,  120, 0,   60, 0,  20, 0,   245, 0,   121;
   720, 0,  720, 0,  360, 0, 120, 0,  2190, 0,  1446, 0,  361;
  5040, 0, 5040, 0, 2520, 0, 840, 0, 20370, 0, 15162, 0, 5047, 0, 841;
  ...

Cf. A000142, A001710, A001715, A180184, A330045, A334157.

# Triangle T(n, k) without the zeros (even k):
W := proc(n, m) local v, s, h; v := 0;
for s from 0 to m do
if 0 = (m - s) mod 4 then
h := (m - s)/4;
v := v + binomial(n - s - 3*h, h)/s!;
end if; end do; n!*v; end proc;
for n1 from 0 to 20 do
seq(W(n1,m1), m1=0..n1); end do; # Petros Hadjicostas, Apr 15 2020

(* Generates the sequence in the data section *)
Table[Expand[t^n*n!*SeriesCoefficient[Series[Exp[t*x]/(1 - x/t - t^4*x^4), {x, 0, 20}], n]], {n, 0, 10}];
a = Table[CoefficientList[Expand[t^n*n!*SeriesCoefficient[Series[Exp[t*x]/(1 - x/t - t^4*x^4), {x, 0, 20}], n]], t], {n, 0, 10}];
Flatten[%]
(* Generates row sums *)
Table[Apply[Plus, CoefficientList[Expand[t^n*n!*SeriesCoefficient[Series[Exp[t*x]/( 1 - x/t - t^4*x^4), {x, 0, 20}], n]], t]], {n, 0, 10}];

T(n,k) = [t^k] (t^n * n! * ([x^n] p(x,t))), where p(x,t) = exp(t*x)/(1 - x/t - t^4*x^4).
From Petros Hadjicostas, Apr 15 2020: (Start)
Sum_{n,k >= 0} (T(n,k)/n!) * (x/t)^n * t^k = p(x,t).
T(n,0) = n! = A000142(n) for n >= 0; T(n,2) = n! for n >= 1; T(n,4) = n!/2 = A001710(n) for n >= 2; T(n,6) = n!/6 = A001715(n) for n >= 3.
T(n,2*m) = n! * Sum_{s = 0..m with 4|(m-s)} binomial(n - s - 3*(m-s)/4, (m-s)/4)/s! for n >= 0 and 0 <= m <= n.
T(n,2*n) = A330045(n) for n >= 0. (End)

Various sections edited by Petros Hadjicostas, Apr 13 2020

A249619 Triangle T(m,n) = number of permutations of a multiset with m elements and signature corresponding to n-th integer partition (A194602).

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 24, 12, 4, 6, 1, 120, 60, 20, 30, 5, 10, 1, 720, 360, 120, 180, 30, 60, 6, 90, 15, 20, 1, 5040, 2520, 840, 1260, 210, 420, 42, 630, 105, 140, 7, 210, 21, 35, 1, 40320, 20160, 6720, 10080, 1680, 3360, 336, 5040, 840, 1120, 56

Offset: 0

Tilman Piesk, Nov 04 2014

This triangle shows the same numbers in each row as A036038 and A078760 (the multinomial coefficients), but in this arrangement the multisets in column n correspond to the n-th integer partition in the infinite order defined by A194602.
Row lengths: A000041 (partition numbers), Row sums: A005651
Columns: 0: A000142 (factorials), 1: A001710, 2: A001715, 3: A133799, 4: A001720, 6: A001725, 10: A001730, 14: A049388
Last in row: end-2: A037955 after 1 term mismatch, end-1: A001405, end: A000012
The rightmost columns form the triangle A173333:
  n       0      1     2     4    6  10  14  21      (A000041(1,2,3...)-1)
m
1         1
2         2      1
3         6      3     1
4        24     12     4     1
5       120     60    20     5    1
6       720    360   120    30    6   1
7      5040   2520   840   210   42   7   1
8     40320  20160  6720  1680  336  56   8   1
A249620 shows the number of partitions of the same multisets. A187783 shows the number of permutations of special multisets.

			Triangle begins:
  n     0    1    2    3   4   5  6   7   8   9 10
m
0       1
1       1
2       2    1
3       6    3    1
4      24   12    4    6   1
5     120   60   20   30   5  10  1
6     720  360  120  180  30  60  6  90  15  20  1

Tilman Piesk, Triangle rows m=0..25, flattened
Tilman Piesk, Illustration of the multisets for m=0..6

Cf. A036038, A078760, A005651, A005651, A249620, A000041, A173333.

A303613 a(n) = [x^n] (1/6 * Sum_{k=0..n} (k+3)!*x^k)^(1/2).

Original entry on oeis.org

1, 2, 8, 44, 300, 2408, 22056, 225824, 2547352, 31322640, 416442016, 5948450912, 90815006528, 1475513165184, 25419300628160, 462839142264960, 8882065110092000, 179190173518717120, 3791700201187720960, 83977684252758211200, 1942984028971973710720

Offset: 0

Seiichi Manyama, Apr 27 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..447

Cf. A001715, A303568.

N=66; x='x+O('x^N); Vec((1/3!*sum(k=0, N, (k+3)!*x^k))^(1/2))

A166553 Triangle read by rows: T(n, k) = [x^k]( (n+2)!(3EulerE(n, x+1) - EulerE(n, x))/4 ).

Original entry on oeis.org

1, 3, 3, 0, 24, 12, -30, 0, 180, 60, 0, -720, 0, 1440, 360, 2520, 0, -12600, 0, 12600, 2520, 0, 120960, 0, -201600, 0, 120960, 20160, -771120, 0, 3810240, 0, -3175200, 0, 1270080, 181440, 0, -61689600, 0, 101606400, 0, -50803200, 0, 14515200, 1814400

Offset: 0

Roger L. Bagula, Dec 12 2010

I think the rows are indexed by t = 0, 1, 2, ..., and in each row we expand the polynomial in powers of x. - N. J. A. Sloane, Dec 14 2010

Former name: Triangle read by rows: expansion of p(x,t) = exp(x*t)*(3*exp(t) - 1)/(exp(t) + 1), with coefficient of x^n scaled by multiplication by (n!*(n + 2)!/4). - G. C. Greubel, Nov 30 2024

			Triangle begins as:
        1;
        3,      3;
        0,     24,      12;
      -30,      0,     180,      60;
        0,   -720,       0,    1440,      360;
     2520,      0,  -12600,       0,    12600,   2520;
        0, 120960,       0, -201600,        0, 120960,   20160;
  -771120,      0, 3810240,       0, -3175200,      0, 1270080, 181440;

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

Cf. A001715, A005990, A028296.

m:= 13;
R:=PowerSeriesRing(Integers(), m+1);
EulerE:= func< n | (2^(n+1)/(n+1))*( Evaluate(BernoulliPolynomial(n+1), 1/2) - 2^(n+1)*Evaluate(BernoulliPolynomial(n+1), 1/4) ) >;
f:= func< n,x | (Factorial(n+2)/2)*( 3*x^n - 2*(&+[ Binomial(n,j)*(EulerE(j)/2^j)*(x - 1/2)^(n-j): j in [0..n]]) ) >;
A166553:= func< n,k | Coefficient(R!( f(n,x) ), k) >;
[A166553(n,k): k in [0..n], n in [0..m]]; // G. C. Greubel, Nov 30 2024

(* first program *)
p[t_]= Exp[x*t](3*Exp[t] - 1)/(Exp[t] + 1);
With[{m=12}, Table[(n!*(n+2)!/2)*CoefficientList[SeriesCoefficient[ Series[p[t], {t,0,m+1}], n], x], {n,0,m}]]//Flatten
(* Second program *)
f[n_, x_]:= (n+2)!*(3*EulerE[n, x+1] - EulerE[n, x])/4;
A166553[n_, k_]:= Coefficient[Series[f[n, x], {x,0,n}], x, k];
Table[A166553[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 30 2024 *)

def f(n,x): return (factorial(n+2)/2)*( 3*x^n - 2*sum( binomial(n,j)*euler_number(j)*(x-1/2)^(n-j)/2^j for j in range(n+1)) )
def A166553(n,k): return ( f(n,x) ).series(x,n+1).list()[k]
print(flatten([[A166553(n,k) for k in range(n+1)] for n in range(14)])) # G. C. Greubel, Nov 30 2024

T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (n!*(n+2)!/2) * [t^n]( exp(x*t)*(3*exp(t) - 1)/(exp(t) + 1) ).
From G. C. Greubel, Nov 30 2024: (Start)
T(n, k) = [x^k]( (n+2)!*(3*EulerE(n, x+1) - EulerE(n, x))/4 ).
T(n, k) = [x^k]( (1/2)*(n+2)!*( 3*x^n - 2*Sum_{j=0..n} binomial(n,j)*(EulerE(j)/2^j)*(x - 1/2)^(n-j) ) ).
T(n, n) = 3*A001715(n+2) = (n+2)!/2.
T(n, n-1) = 3*A005990(n+1). (End)

I rewrote the definition. - N. J. A. Sloane, Dec 14 2010

New name by G. C. Greubel, Nov 30 2024

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