A007526 -id:A007526 - cp's OEIS frontend

A337001 a(n) = n! * Sum_{k=0..n} k^3 / k!.

0, 1, 10, 57, 292, 1585, 9726, 68425, 547912, 4931937, 49320370, 542525401, 6510306540, 84633987217, 1184875823782, 17773137360105, 284370197765776, 4834293362023105, 87017280516421722, 1653328329812019577, 33066566596240399540, 694397898521048399601

Offset: 0

Ilya Gutkovskiy, Aug 10 2020

nonn

Exponential convolution of cubes (A000578) and factorial numbers (A000142).

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A000142, A000522, A000578, A007526, A030297, A256016, A337002.

Row sums of A121682.

Table[n! Sum[k^3/k!, {k, 0, n}], {n, 0, 21}]
nmax = 21; CoefficientList[Series[x (1 + 3 x + x^2) Exp[x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 0; a[n_] := a[n] = n (n^2 + a[n - 1]); Table[a[n], {n, 0, 21}]

a(n) = n! * sum(k=0, n, k^3/k!); \\ Michel Marcus, Aug 12 2020

E.g.f.: x * (1 + 3*x + x^2) * exp(x) / (1 - x).
a(0) = 0; a(n) = n * (n^2 + a(n-1)).
a(n) ~ 5*exp(1)*n!. - Vaclav Kotesovec, Jan 13 2024

A093964 a(n) = Sum_{k=1..n} kk!C(n,k).

Original entry on oeis.org

0, 1, 6, 33, 196, 1305, 9786, 82201, 767208, 7891281, 88776910, 1085051121, 14322674796, 203121569833, 3080677142466, 49764784609065, 853110593298256, 15469738758475041, 295858753755835158, 5951981987323272001, 125652953065713520020, 2777591594084193600441

Offset: 0

Ralf Stephan, Apr 20 2004

nonn

Limit to which the columns of array A093966 converge.
Number of objects in all permutations of n objects taken 1,2,...,n at a time. Example: a(2)=6 because the permutations of {a,b} taken 1 and 2 at a time are: a,b,ab and ba, containing altogether 1+1+2+2=6 objects. a(n)=Sum(k*A008279(n,k),k=1..n). - Emeric Deutsch, Aug 16 2006
The number of sequences -where each member is an element in a set consisting of n elements- such that the last member is a repetition of a former member. Example: Set of possible members: {l,r}. Sequences such that the last member is a repetition of a former member: l,l; r,r; l,r,l; l,r,r; r,l,l; r,l,r. a(n)=Sum(k*A008279(n,k),k=1..n). [From Franz Fritsche (ff(AT)simple-line.de), Feb 22 2009]
The total number of elements in all ascending runs (including runs of length 1) over all permutations of {1,2,...,n}.  a(2) = 6 because in the permutations [1,2] and [2,1] there are 4 runs of length 1 and 1 run of length 2. a(n) = Sum_{k>=1} A132159(n,k)*k. - Geoffrey Critzer, Feb 24 2014

			G.f. = x + 6*x^2 + 33*x^3 + 196*x^4 + 1305*x^5 + 9786*x^6 + 82201*x^7 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A000522, A001339, A007526, A008279.

Row n=2 of A210472. - Alois P. Heinz, Jan 23 2013

[0] cat [n le 2 select 6^(n-1) else n*((n+1)*Self(n-1) - (n-1)*Self(n-2))/(n-1): n in [1..30]]; // G. C. Greubel, Dec 29 2021

seq(add(k*n!/(n-k)!,k=1..n),n=0..20); # Emeric Deutsch, Aug 16 2006
# second Maple program:
a:= proc(n) a(n):=`if`(n<2, n, n*((n+1)/(n-1)*a(n-1)-a(n-2))) end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 21 2013

nn=21;Range[0,nn]!CoefficientList[Series[D[Exp[y x]/(1-x)^2,y]/.y->1,{x,0,nn}],x] (* Geoffrey Critzer, Feb 24 2014 *)

PARI
```
a(n)=sum(k=1,n,k*k!*binomial(n,k))
```

[factorial(n)*( x*exp(x)/(1-x)^2 ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Dec 29 2021

E.g.f.: x*exp(x)/(1-x)^2. - Vladeta Jovovic, Apr 24 2004
a(n) = 1 + (n-1)*floor(e*n!) = 1 + (n-1)*A000522(n) = A000522(n+1) - 2*A000522(n) = A001339(n) - A000522(n). - Henry Bottomley, Dec 22 2008
a(n) = n if n < 2, a(n) = n*((n+1)/(n-1)*a(n-1) - a(n-2)) for n >= 2. - Alois P. Heinz, Jan 21 2013
E.g.f.: x*(1- 12*x/(Q(0)+6*x-3*x^2))/(1-x)^2, where Q(k) = 2*(4*k+1)*(32*k^2+16*k+x^2-6) - x^4*(4*k-1)*(4*k+7)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Nov 18 2013
G.f.: conjecture: T(0)/x - 1/x, where T(k) = 1 - x^2*(k+1)^2/(x^2*(k+1)^2 - (1 - 2*x*(k+1))*(1 - 2*x*(k+2))/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 18 2013
a(n) = n*a(n-1) + A007526(n), a(0) = 0. - David M. Cerna, May 12 2014

a(0) inserted by Alois P. Heinz, Jan 21 2013

A038154 a(n) = n! * Sum_{k=0..n-2} 1/k!.

Original entry on oeis.org

0, 0, 2, 12, 60, 320, 1950, 13692, 109592, 986400, 9864090, 108505100, 1302061332, 16926797472, 236975164790, 3554627472060, 56874039553200, 966858672404672, 17403456103284402, 330665665962403980, 6613313319248079980, 138879579704209680000

Offset: 0

N. J. A. Sloane

nonn

The number of rank-orderings of (>=2)-element subsets of an n-set. (Counts nontrivial votes in a rank-ordering voting system.) E.g., a(5) = 320 = 120+120+60+20 because of 5-, 4-, 3- and 2-element subsets. - Warren D. Smith, Jul 06 2005
a(n) is the number of simple cycles through a vertex of the complete graph K_(n+1) on n+1 vertices [Hassani]. For example, in the complete graph K_4 with vertex set {A,B,C,D} there are a(3) = 12 simple cycles at the vertex A, namely the six 3-cycles ABCA, ABDA, ACBA, ACDA, ADBA and ADCA and the six 4-cycles ABCDA, ABDCA, ACBDA, ACDBA, ADBCA and ADCBA. The sum of the lengths of the cycles at a vertex of K_n is equal to A141834(n). - Peter Bala, Jul 09 2008
See A000522 for the number of paths between a pair of distinct vertices of K_n. - Peter Bala, Jul 09 2008
a(n) = n*a(n-1) + A000217(n-1), where A000217(n) is the n-th triangular number. - Gary Detlefs, May 20 2010

			0=1*0+0, 2=2*0+2, 12=3*2+6, 60=4*12+12, 320 = 5*60+20, ... - _Gary Detlefs_, May 20 2010

Mehdi Hassani, Counting and computing by e, arXiv:math/0606613 [math.CO], 2006.
J. Sawada and A. Williams, Successor rules for flipping pancakes and burnt pancakes, Preprint 2015.
Index entries for sequences related to factorial numbers

Cf. A000522, A038155, A141834.

Table[n!Sum[1/k!,{k,0,n-2}],{n,0,30}] (* Harvey P. Dale, Jun 04 2012 *)

main(size)=my(k); vector(size,n,(n-1)!*sum(k=0,n-3,1/k!)); \\ Anders Hellström, Jul 14 2015

a(n) = A007526(n) - n.
a(n) = floor(n!*exp(1))-n-1, n>0. - Vladeta Jovovic, Aug 25 2001
a(n) = n*a(n-1) + n*(n-1), for n>=3, a(2)=2 and a(3) = 12. - Ian Myers, Jul 19 2012
a(n) = A000522(n-2) * n*(n-1). - Doug Bell, Jun 30 2015
E.g.f.: exp(x)*x^2/(1 - x). - Ilya Gutkovskiy, Jan 26 2017
a(n) = 2*A038155(n). - Alois P. Heinz, Jan 26 2017

A271705 Triangle read by rows, T(n,k) = Sum_{j=0..n} C(n,j)*L(j,k), L the unsigned Lah numbers A271703, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 15, 9, 1, 1, 64, 66, 16, 1, 1, 325, 490, 190, 25, 1, 1, 1956, 3915, 2120, 435, 36, 1, 1, 13699, 34251, 23975, 6755, 861, 49, 1, 1, 109600, 328804, 283136, 101990, 17696, 1540, 64, 1, 1, 986409, 3452436, 3534636, 1554966, 342846, 40404, 2556, 81, 1

Offset: 0

Peter Luschny, Apr 14 2016

This is the Sheffer (aka exponential Riordan) matrix T = P*L = A007318*A271703 = (exp(x), x/(1-x)). Note that P = A007318 is Sheffer (exp(t), t) (of the Appell type). The Sheffer a-sequence is [1,1,repeat(0)] and the z-sequence has e.g.f. (x/(1+x))*(1 - exp(-x/(1+x)) given in A288869 / A000027. Because the column k=0 has only entries 1, the z-sequence gives fractional representations of 1. See A288869. - Wolfdieter Lang, Jun 20 2017

			Triangle starts:
  1;
  1,    1;
  1,    4,    1;
  1,   15,    9,    1;
  1,   64,   66,   16,   1;
  1,  325,  490,  190,  25,  1;
  1, 1956, 3915, 2120, 435, 36, 1;
  ...
Recurrence: T(3, 2) = (3/2)*4 + 3*1 = 9. - _Wolfdieter Lang_, Jun 20 2017

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.

Cf. A000290 (diag n, n-1), A062392 (diag n, n-2).
Cf. A007526 (col. 1), A134432 (col. 2).
Cf. A052844 (row sums), A059110 (matrix inverse).
Cf. A007318, A271703, A288869.

B:=Binomial;
A271705:= func< n,k | k eq 0 select 1 else (&+[B(n, j+k)*B(j+k, k)*B(j+k-1, k-1)*Factorial(j): j in [0..n-k]]) >;
[A271705(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 09 2022

L := (n,k) -> `if`(k<0 or k>n,0,(n-k)!*binomial(n,n-k)*binomial(n-1,n-k)):
T := (n,k) -> add(L(j,k)*binomial(-j-1,-n-1)*(-1)^(n-j), j=0..n):
seq(seq(T(n,k), k=0..n), n=0..9);

T[n_, k_]:= If[k==0, 1, Sum[((k*j!)/(j+k))*Binomial[n, j+k]*Binomial[j+k, k]^2, {j,0,n-k}]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 09 2022 *)

b=binomial
def A271705(n,k): return 1 if (k==0) else sum(factorial(j-k)*b(n, j)*b(j, k)*b(j-1, k-1) for j in (k..n))
flatten([[A271705(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jan 09 2022

From Wolfdieter Lang, Jun 20 2017: (Start)
T(n, k) = Sum_{m=k..n} A007318(n, m)*A271703(m, k), n >= k >= 0, and 0 for k < m. See also the name.
E.g.f. of column k: exp(x)*(x/(1-x))^k/k! (Sheffer property), k >= 0.
E.g.f. of triangle (or row polynomials in x): exp(z)*exp(x*z/(1-z)).
Recurrence for T(n, k), k >= 1, with T(n, 0) = 1, T(n, k) = 0 if n < k: T(n, k) = (n/k)*T(n-1, k-1) + n*T(n-1, k), n >= 1, k = 1..n. (From the a-sequence with column k=0 as input.) (End)
T(n, k) = Sum_{j=0..n-k} j!*binomial(n, j+k)*binomial(j+k, k)*binomial(j+k-1, k-1) with T(n, 0) = 1. - G. C. Greubel, Jan 09 2022
From Natalia L. Skirrow, Jun 11 2025: (Start)
T(n, k) = C(n, k)*hypergeom([k-n, k], [], -1), which equals C(n, k)*A143409(n-k, k-1) for k>0.
By the saddle point method upon the e.g.f., n-th row polynomial converges with n (for all y) to n^n*exp(2*sqrt(n*y) - n - y/2 + 1)/sqrt(2*sqrt(n/y)); as such, the n-th row's expectation is ~ sqrt(n)-1/4 and the n-th row's variance is ~ (sqrt(n)-1)/2. (End)

A103519 a(1) = 1, a(n) = Sum_{k=1..n} a(n-1) + k.

Original entry on oeis.org

1, 5, 21, 94, 485, 2931, 20545, 164396, 1479609, 14796145, 162757661, 1953092010, 25390196221, 355462747199, 5331941208105, 85311059329816, 1450288008607025, 26105184154926621, 495998498943605989, 9919969978872119990

Offset: 1

Amarnath Murthy, Feb 10 2005

Eigensequence of a triangle with the natural numbers (1, 2, 3, ...) as the right border, the triangular series (1, 3, 6, ...) as the left border; and the rest zeros. - Gary W. Adamson, Aug 01 2016

			a(2) = 2 + 3 = 5, a(3) = 6 + 7 + 8 = 21, a(4) = 22 + 23 + 24 + 25.

Harvey P. Dale, Table of n, a(n) for n = 1..449

Cf. A007526, A103520.

a[1]:=1: for n from 2 to 20 do a[n]:=n*a[n-1]+(1/2)*n*(n+1) end do: seq(a[n], n=1..20); # Emeric Deutsch, Mar 16 2008

RecurrenceTable[{a[1]==1,a[n]==n*a[n-1]+(n(n+1))/2},a,{n,20}] (* Harvey P. Dale, Nov 05 2013 *)

{ t(n) = n*(n+1)/2 }
{ a(n) = sum(i=0,n,n!/(n-i)!*t(n-i)) } \\ Max Alekseyev, Feb 14 2005

{ t(n) = n*(n+1)/2 }
{ a(n) = 3*t(n-1)*floor((n-2)!*exp(1))+n } \\ Max Alekseyev, Feb 14 2005

a(n+1) = k*(k+1)/2 - a(n)*(a(n)+1)/2, where k = a(n) + n + 1.
a(n) = Sum_{i=0..n} (n!/(n-i)!) * (n-i)(n-i+1)/2 = Sum_{i=0..n} (n!/(n-i)!) * A000217(n-i). For n > 2, a(n) = (3*n*(n-1)/2)*floor((n-2)!*e) + n, where e=exp(1). - Max Alekseyev, Feb 14 2005
a(n) = n*a(n-1) + n*(n+1)/2. - Emeric Deutsch, Mar 16 2008
a(n) ~ 3*sqrt(Pi/2)*exp(1)*n^n*sqrt(n)/exp(n). - Ilya Gutkovskiy, Aug 02 2016
E.g.f.: x * (1+x/2) * exp(x) / (1-x). - Seiichi Manyama, Dec 31 2023

More terms from Max Alekseyev, Feb 14 2005

Name clarified by Seiichi Manyama, Dec 31 2023

A337002 a(n) = n! * Sum_{k=0..n} k^4 / k!.

Original entry on oeis.org

0, 1, 18, 135, 796, 4605, 28926, 204883, 1643160, 14795001, 147960010, 1627574751, 19530917748, 253901959285, 3554627468406, 53319412076715, 853110593292976, 14502880086064113, 261051841549259010, 4959984989436051511, 99199699788721190220

Offset: 0

Ilya Gutkovskiy, Aug 10 2020

nonn

Exponential convolution of fourth powers (A000583) and factorial numbers (A000142).

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A000142, A000522, A000583, A007526, A030297, A256016, A337001.

Table[n! Sum[k^4/k!, {k, 0, n}], {n, 0, 20}]
nmax = 20; CoefficientList[Series[x (1 + 7 x + 6 x^2 + x^3) Exp[x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 0; a[n_] := a[n] = n (n^3 + a[n - 1]); Table[a[n], {n, 0, 20}]

a(n) = n! * sum(k=0, n, k^4/k!); \\ Michel Marcus, Aug 12 2020

E.g.f.: x * (1 + 7*x + 6*x^2 + x^3) * exp(x) / (1 - x).
a(0) = 0; a(n) = n * (n^3 + a(n-1)).
a(n) ~ 15*exp(1)*n!. - Vaclav Kotesovec, Jan 13 2024

A111063 a(0) = 1; a(n) = (n-1)*a(n-1) + n.

Original entry on oeis.org

1, 1, 3, 9, 31, 129, 651, 3913, 27399, 219201, 1972819, 19728201, 217010223, 2604122689, 33853594971, 473950329609, 7109254944151, 113748079106433, 1933717344809379, 34806912206568841, 661331331924807999, 13226626638496160001, 277759159408419360043

Offset: 0

N. J. A. Sloane, Oct 08 2005

From Frank Ruskey, Nov 24 2009: (Start)
If the initial 1 were deleted, the recurrence relation becomes a(n) = n+1+n*a(n-1) with a(0) = 1. Furthermore, it can then be shown that a(n) is the number of nonempty subsets of binary strings with n 1's and 2 0's that are closed under the operation of replacing the leftmost 01 with 10. Taking the maximal elements under this relation,
a(2) = 9 = |{0011},{0101},{1001},{1010},{1100},{0110}, {0110,1001},{0101,0110},{0011,0110}|.
We also have the e.g.f. (1+x)/(1-x) e^x and the formula a(n) = -1 + 2*n!*sum_{j=0..n} 1/j!. (End)
a(n+1) = sum of n-th row in triangle A245334. - Reinhard Zumkeller, Aug 30 2014 [A-number corrected by N. J. A. Sloane, May 03 2017]
Eigensequence of triangle with (1, 2, 3, ...) as the right and left borders and the rest zeros. - Gary W. Adamson, Aug 01 2016
The following remarks apply to the sequence without the initial term a(0) = 1: For k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k. It follows that for each k the sequence a(n) taken modulo k is periodic with period dividing k. For example, modulo 10 the sequence becomes 1, 3, 9, 1, 9, 1, 3, 9, 1, 9, ... with period 5. Cf. A000522. - Peter Bala, Nov 19 2017

F. Drewes et al., Tight Bounds for Cut-Operations on Deterministic Finite Automata, in Lecture Notes in Computer Science, Volume 9288 2015, Machines, Computations, and Universality, 7th International Conference, MCU 2015, Famagusta, North Cyprus, September 9-11, 2015, Editors: Jerome Durand-Lose, Benedek Nagy, ISBN: 978-3-319-23110-5 (Print) 978-3-319-23111-2 (Online). ["In the On-Line Encyclopedia of Integer Sequences (OEIS) this matches the sequence A111063."]

Reinhard Zumkeller, Table of n, a(n) for n = 0..400

Cf. A000522, A007526, A030297, A245334, A240993.

a111063 n = a111063_list !! n
a111063_list = 1 : zipWith (+) [1..] (zipWith (*) [0..] a111063_list)
-- Reinhard Zumkeller, Aug 30 2014

a:=proc(n) option remember; if n=0 then RETURN(1); fi; (n-1)*a(n-1)+n; end;
# Alternatively:
a := n -> `if`(n=0, 1, 2*exp(1)*GAMMA(n, 1) - 1):
seq(simplify(a(n)), n=0..22); # Peter Luschny, Nov 21 2017

FoldList[#1*#2 + #2 + 1 &, 1, Range[21]] (* Robert G. Wilson v, Jul 07 2012 *)

a(n+1) = Sum_{k=0..2*n} C(n,floor(k/2))(n-floor(k/2))!. - Paul Barry, May 04 2007
a(n) = A030297(n)/n, n>0.
a(n) = A007526(n) + A000522(n). - Gary Detlefs, Jun 10 2010
a(n) = 2*floor(e*n!) - 1, n>1. - Gary Detlefs, Jun 10 2010
E.g.f.: exp(x)*(1+x)/(1-x), - N. J. A. Sloane, May 03 2017
a(n) ~ 2*sqrt(2*Pi)*exp(1)*n^n*sqrt(n)/exp(n). - Ilya Gutkovskiy, Aug 02 2016
a(n) = 2*exp(1)*GAMMA(n, 1) - 1 for n>=1. - Peter Luschny, Nov 21 2017

A134597 a(n) gives the maximal value of A075053(m) for any n-digit number m.

Original entry on oeis.org

1, 4, 11, 31, 106, 402, 1953

Offset: 1

N. J. A. Sloane, Jan 25 2008

In A075053(m), the primes obtained as permutations of digits of m are counted several times if they can be obtained in several different ways. See sequence A076730 which uses A039993 instead, i.e., counting only different primes. - M. F. Hasler, Mar 11 2014

The original data given for n = 3, 4, 5 was erroneously A007526(n). - Up to n = 6, a(n) = A076730(n), but the two will differ not later than for n = 10, where A134596(10) = 1123456789 gives a(10) >= 398100 = A075053(1123456789) > A039993(1123456789) = 362451 = A076730(10). The difference arises because each prime containing a single '1' will be counted twice by A075053, but only once by A039993. - M. F. Hasler, Oct 14 2019

			From _M. F. Hasler_, Oct 14 2019: (Start)
a(2) = 4 = A075053(37), because from 37 one can obtain the primes {3, 7, 37, 73}, and there is obviously no 2-digit number which could give more primes.
a(3) = 11 = A075053(137), because from 137 one can obtain the primes {3, 7, 13, 17, 31, 37, 71, 73, 137, 173, 317}, and no 3-digit number yields more.
a(4) = 31 = A075053(1379), because from 1379 one can obtain the 31 primes {3, 7, 13, 17, 19, 31, 37, 71, 73, 79, 97, 137, 139, 173, 179, 193, 197, 317, 379, 397, 719, 739, 937, 971, 1973, 3719, 3917, 7193, 9137, 9173, 9371}, and no 4-digit number yields more.
a(5) = 106 = A075053(13679). a(6) = 402 = A075053(123479).
a(7) = 1953 = A075053(1234679). (End)

M. Keith, Integers containing many embedded primes

Cf. A075053, A039993, A134596.

Cf. A239196 for record indices of A075053, A239197 for associated record values.

A134597(n)={my(m=0);forvec(D=vector(n,i,[0,9]), vecsum(D)%3||next;m=max(A075053(fromdigits(D),D),m),1);m} \\ M. F. Hasler, Oct 14 2019

a(n) <= A007526(n), with equality iff n <= 2. [Keith]

a(n) = max { A075053(m); 10^(n-1) <= m < 10^n } >= A076730(n) = max { A039993(m); 10^(n-1) <= m < 10^n }. - M. F. Hasler, Mar 11 2014

Link fixed by Charles R Greathouse IV, Aug 13 2009
Definition corrected by M. F. Hasler, Mar 11 2014
Data corrected and extended by M. F. Hasler, Oct 14 2019

A128195 a(n) = (2n + 1)(a(n - 1) + 2^n) for n >= 1, a(0) = 1.

Original entry on oeis.org

1, 9, 65, 511, 4743, 52525, 683657, 10256775, 174369527, 3313030741, 69573667065, 1600194389599, 40004859842375, 1080131215965309, 31323805263469097, 971037963168557815, 32044252784564570583, 1121548847459764557925, 41497307356011298342553, 1618394986884440655806799

Offset: 0

Peter Luschny, Feb 26 2007

P. Luschny, Variants of Variations.

Cf. A007526 (The number of variations), A128196 (A weighted sum of double factorials), A126062.

a := n -> `if`(n=0,1,(2*n+1)*(a(n-1)+2^n));

a[0] = 1; a[n_] := a[n] = (2*n+1)*(a[n-1] + 2^n); Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Jul 29 2013 *)

a(n) = A126062(2, n), double variations.
a(n) = (2n+1)!/(n! 2^n) Sum(k=0..n, 4^k*k!/(2k)!) [Gottfried Helms]
a(n) = 2^n (2n+1) Sum(k=0..n, Gamma(n+1/2)/Gamma(k+1/2))
a(n) = 2^(n+1) Gamma(n+3/2) Sum(k=0..n, 1/Gamma(k+1/2))
a(n) = A128196(n)*A005408(n)
a(n) = A128196(n+1)-A000079(n+1)
Recursive form:
a(n) = 2^(n+1)*v(n+1/2) with v(x) = if x <= 1 then x else x(v(x-1)+1).
a(n) = (2n+1)*(a(n-1)+2^n), a(0) = 1 [Wolfgang Thumser]
Note: The following constants will be used in the next formulas.
K = (1-exp(1)*Gamma(1/2,1))/Gamma(1/2)
M = sqrt(2)(1+exp(1)(Gamma(1/2)-Gamma(1/2,1)))
Generalized form: For x>0
a(x) = 2^(x+1)(x+1/2)(exp(1) Gamma(x+1/2,1) + K Gamma(x+1/2))
Asymptotic formula:
a(n) ~ 2^(n+5/2)*Gamma(n+3/2)
a(n) ~ (exp(1)+K)*2^(n+1)*(n+1/2)!
a(n) ~ M(2n+1)(2exp(-1)(n-1/(24*n+19/10*1/n)))^n

A326659 T(n,k) = [0=0]; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 15, 18, 6, 1, 64, 132, 96, 24, 1, 325, 980, 1140, 600, 120, 1, 1956, 7830, 12720, 10440, 4320, 720, 1, 13699, 68502, 143850, 162120, 103320, 35280, 5040, 1, 109600, 657608, 1698816, 2447760, 2123520, 1108800, 322560, 40320

Offset: 0

Alois P. Heinz, Sep 12 2019

[] is an Iverson bracket.

			Triangle T(n,k) begins:
  1;
  1,     1;
  1,     4,     2;
  1,    15,    18,      6;
  1,    64,   132,     96,     24;
  1,   325,   980,   1140,    600,    120;
  1,  1956,  7830,  12720,  10440,   4320,   720;
  1, 13699, 68502, 143850, 162120, 103320, 35280, 5040;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Iverson bracket

Columns k=0-2 give: A000012, A007526, 2*A134432(n-1).
Main diagonal gives A000142.
Row sums give A308876.
Cf. A000522, A073474, A093964, A143409, A196347, A271705, A327606.

T:= proc(n, k) option remember;
      `if`(0=0, 1, 0)
    end:
seq(seq(T(n, k), k=0..n), n=0..10);

T[n_ /; n >= 0, k_ /; k >= 0] := T[n, k] = Boole[0 < k <= n]*n*(T[n-1, k-1] + T[n-1, k]) + Boole[k == 0 && n >= 0];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 09 2021 *)

E.g.f. of column k: exp(x)*(x/(1-x))^k.
T(n,k) = k! * A271705(n,k).
T(n,k) = n * A073474(n-1,k-1) for n,k >= 1.
T(n,1) = n * A000522(n-1) for n >= 1.
T(n,2) = n * A093964(n-1) for n >= 1.
Sum_{k=1..n} k * T(n,k) = A327606(n).

cp's OEIS Frontend

A337001 a(n) = n! * Sum_{k=0..n} k^3 / k!.

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A093964 a(n) = Sum_{k=1..n} k*k!*C(n,k).

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A038154 a(n) = n! * Sum_{k=0..n-2} 1/k!.

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A271705 Triangle read by rows, T(n,k) = Sum_{j=0..n} C(n,j)*L(j,k), L the unsigned Lah numbers A271703, for n>=0 and 0<=k<=n.

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A103519 a(1) = 1, a(n) = Sum_{k=1..n} a(n-1) + k.

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A337002 a(n) = n! * Sum_{k=0..n} k^4 / k!.

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A093964 a(n) = Sum_{k=1..n} kk!C(n,k).

A128195 a(n) = (2n + 1)(a(n - 1) + 2^n) for n >= 1, a(0) = 1.