A007526 -id:A007526 - cp's OEIS frontend

A193563 a(0) = 0, a(n) = n^2 * (a(n-1) + 1).

0, 1, 8, 81, 1312, 32825, 1181736, 57905113, 3705927296, 300180111057, 30018011105800, 3632179343801921, 523033825507476768, 88392716510763573961, 17324972436109660496552, 3898118798124673611724425, 997918412319916444601453056

Offset: 0

Meherzad Lahewala, Jul 31 2011

Paolo Xausa, Table of n, a(n) for n = 0..250

Cf. A006040, A007526 (multiply by n instead of n^2), A180255.

seq(n!^2*add(1/k!^2,k=0..n-1),n=0..16);   # Mark van Hoeij, May 13 2013

FoldList[#2^2*(# + 1) &, Range[0, 20]] (* Paolo Xausa, Jun 18 2025 *)

a=[0];for(n=1,20,a=concat(a,(a[#a]+1)*n^2));a \\ Charles R Greathouse IV, Jul 31 2011

From Seiichi Manyama, Jan 05 2024: (Start)
a(n) = (n!)^2 * Sum_{k=0..n} (k/k!)^2.
a(n) = n^2 * A006040(n-1) for n > 0. (End)
a(n) = Sum_{k=1..n} (k!*binomial(n,k))^2. - Ridouane Oudra, Jun 14 2025
a(n) = n^2 + BesselI(0,2)*(n!)^2 - n^2*hypergeom([1], [n, n], 1) for n > 0. - Stefano Spezia, Jun 14 2025

A285268 Triangle read by rows: T(m,n) = Sum_{i=1..n} P(m,i) where P(m,n) = m!/(m-n)! is the number of permutations of m items taken n at a time, for 1 <= n <= m.

Original entry on oeis.org

1, 2, 4, 3, 9, 15, 4, 16, 40, 64, 5, 25, 85, 205, 325, 6, 36, 156, 516, 1236, 1956, 7, 49, 259, 1099, 3619, 8659, 13699, 8, 64, 400, 2080, 8800, 28960, 69280, 109600, 9, 81, 585, 3609, 18729, 79209, 260649, 623529, 986409, 10, 100, 820, 5860, 36100, 187300, 792100, 2606500, 6235300, 9864100

Offset: 1

Rick Nungester, Apr 15 2017

T(m, n) is the total number of ordered sets of size 1 to n that can be created from m distinct items. For example, for 4 items taken 1 to 3 at a time, there are P(4, 1) + P(4, 2) + P(4, 3) = 4 + 12 + 24 = 40 total sets: 1, 12, 123, 124, 13, 132, 134, 14, 142, 143, 2, 21, 213, 214, 23, 231, 234, 24, 241, 243, 3, 31, 312, 314, 32, 321, 324, 34, 341, 342, 4, 41, 412, 413, 42, 421, 423, 43, 431, 432.
T(m, n) is the total number of tests in a software program that generates all P(m, n) possible solutions to a problem, allowing early-termination testing on each partial permutation, and not doing any such early termination. For example, from a deck of 52 cards, evaluate all possible 5-card deals as each card is dealt. T(52, 5) = 318507904 total evaluations.
T(m, n) counts the numbers with <= n distinct nonzero digits in base m+1. - M. F. Hasler, Oct 10 2019

			Triangle begins:
  1;
  2,  4;
  3,  9,  15;
  4, 16,  40,   64;
  5, 25,  85,  205,   325;
  6, 36, 156,  516,  1236,  1956;
  7, 49, 259, 1099,  3619,  8659,  13699;
  8, 64, 400, 2080,  8800, 28960,  69280, 109600;
  9, 81, 585, 3609, 18729, 79209, 260649, 623529, 986409;
  ...

Rick Nungester, Table of n, a(n) for n = 1..20100

Mirror of triangle A121662.
Row sums give A030297.
Diagonals (1..4): A007526 (less the initial 0), A038156 (less the initial 0, 0), A224869 (less the initial -1, 0), A079750 (less the initial 0).
Columns (1..3): A000027, A000290 (less the initial 0, 1), A053698 (less the initial 1, 4).

SumPermuteTriangle := proc(M)
  local m;
  for m from 1 to M do print(seq(add(m!/(m-k)!, k=1..n), n=1..m)) od;
end:
SumPermuteTriangle(10);
# second Maple program:
T:= proc(n, k) option remember;
     `if`(k<1, 0, T(n-1, k-1)*n+n)
    end:
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Jun 26 2022

Table[Sum[m!/(m - i)!, {i, n}], {m, 9}, {n, m}] // Flatten (* Michael De Vlieger, Apr 22 2017 *)
(* Sum-free code *)
b[j_] = If[j==0, 0, Floor[j! E - 1]]; T[m_,n_] = b[m] - m! b[m-n]/(m-n)!; Table[T[m, n],{m, 24},{n, m}]//Flatten
(* Manfred Boergens, Jun 22 2022 *)

A285268(m,n,s=m-n+1)={for(k=m-n+2,m,s=(s+1)*k);s} \\ Much faster than sum(k=1,n,m!\(m-k)!), e.g., factor 6 for m=1..99, factor 57 for m=1..199.
apply( A285268_row(m)=vector(m,n,A285268(m,n)), [1..9]) \\ M. F. Hasler, Oct 10 2019

T(n, k) = {exp(1)*(incgam(n+1, 1) - incgam(n-k, 1)*(n-k)*n!/(n-k)!) - 1;}
  apply(Trow(n) = vector(n, k, round(T(n, k))), [1..10]) \\ Adjust the realprecision if needed. Peter Luschny, Oct 10 2019

T(m, n) = Sum_{k=1..n} m!/(m-k)! = m*(1 + (m-1)*(1 + (m-2)*(1 + ... + m-n+1)...)), cf. PARI code. - M. F. Hasler, Oct 10 2019
T(n, k) = exp(1)*(Gamma(n+1, 1) - Gamma(n-k, 1)*(n-k)*n!/(n-k)!) - 1. - Peter Luschny, Oct 10 2019
Sum-free and Gamma-free formula: T(m, n) = b(m) - m!*b(m-n)/(m-n)! where b(0)=0, b(j)=floor(j!*e-1) for j>0. - Manfred Boergens, Jun 22 2022

A357479 a(n) = (n!/6) * Sum_{k=0..n-3} 1/k!.

Original entry on oeis.org

0, 0, 0, 1, 8, 50, 320, 2275, 18256, 164388, 1644000, 18084165, 217010200, 2821132886, 39495860768, 592437911975, 9479006592160, 161143112067400, 2900576017214016, 55110944327067273, 1102218886541346600, 23146596617368279930, 509225125582102160000

Offset: 0

Seiichi Manyama, Sep 30 2022

Seiichi Manyama, Table of n, a(n) for n = 0..450

Column k=3 of A073107.
Cf. A000522, A007526, A038155, A357480.
Cf. A000292, A000449.

Table[n!/6 Sum[1/k!,{k,0,n-3}],{n,0,30}] (* Harvey P. Dale, Apr 02 2023 *)

PARI
```
a(n) = n!/6*sum(k=0, n-3, 1/k!);
```

a(n) = n!*sum(k=0, n, binomial(k, 3)/k!);

my(N=30, x='x+O('x^N)); concat([0, 0, 0], Vec(serlaplace(x^3/6*exp(x)/(1-x))))

my(N=30, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=3, N, k!*x^k/(1-x)^(k+1))/6))

a(n) = n! * Sum_{k=0..n} binomial(k,3)/k!.
a(0) = 0; a(n) = n * a(n-1) + binomial(n,3).
E.g.f.: x^3/6 * exp(x)/(1-x).
G.f.: (1/6) * Sum_{k>=3} k! * x^k/(1-x)^(k+1).

A357480 a(n) = (n!/24) * Sum_{k=0..n-4} 1/k!.

Original entry on oeis.org

0, 0, 0, 0, 1, 10, 75, 560, 4550, 41076, 410970, 4521000, 54252495, 705283150, 9873965101, 148109477880, 2369751647900, 40285778016680, 725144004303300, 13777736081766576, 275554721635336365, 5786649154342069650, 127306281395525539615, 2928044472097087420000

Offset: 0

Seiichi Manyama, Sep 30 2022

Seiichi Manyama, Table of n, a(n) for n = 0..450

Column k=4 of A073107.
Cf. A000522, A007526, A038155, A357479.
Cf. A000332, A000475.

PARI
```
a(n) = n!/24*sum(k=0, n-4, 1/k!);
```

a(n) = n!*sum(k=0, n, binomial(k, 4)/k!);

my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(serlaplace(x^4/24*exp(x)/(1-x))))

my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=4, N, k!*x^k/(1-x)^(k+1))/24))

a(n) = n! * Sum_{k=0..n} binomial(k,4)/k!.
a(0) = 0; a(n) = n * a(n-1) + binomial(n,4).
E.g.f.: x^4/24 * exp(x)/(1-x).
G.f.: (1/24) * Sum_{k>=4} k! * x^k/(1-x)^(k+1).

A368576 a(n) = n! * Sum_{k=0..n} binomial(k+4,5) / k!.

Original entry on oeis.org

0, 1, 8, 45, 236, 1306, 8088, 57078, 457416, 4118031, 41182312, 453008435, 5436105588, 70669378832, 989371312216, 14840569694868, 237449115133392, 4036634957288013, 72659429231210568, 1380529155393034441, 27610583107860731324, 579822245265075410934

Offset: 0

Seiichi Manyama, Dec 31 2023

Cf. A007526, A103519, A368574, A368575.

Cf. A000389.

my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(x*sum(k=0, 4, binomial(4, k)*x^k/(k+1)!)*exp(x)/(1-x))))

a(0) = 0; a(n) = n*a(n-1) + binomial(n+4,5).

E.g.f.: x * (1+2*x+x^2+x^3/6+x^4/120) * exp(x) / (1-x).

A371898 Triangle read by rows: T(n, k) = n * k * (T(n-1, k-1) + T(n-1, k)) for k > 0 with initial values T(n, 0) = 1 and T(i, j) = 0 for j > i.

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 1, 15, 48, 36, 1, 64, 504, 1008, 576, 1, 325, 5680, 22680, 31680, 14400, 1, 1956, 72060, 510480, 1304640, 1382400, 518400, 1, 13699, 1036224, 12233340, 50823360, 94046400, 79833600, 25401600, 1, 109600, 16798768, 318469536, 2017814400, 5794790400, 8346240000, 5893171200, 1625702400

Offset: 0

Werner Schulte, Apr 11 2024

			Lower triangular array starts:
n\k :  0      1        2         3         4         5         6         7
==========================================================================
  0 :  1
  1 :  1      1
  2 :  1      4        4
  3 :  1     15       48        36
  4 :  1     64      504      1008       576
  5 :  1    325     5680     22680     31680     14400
  6 :  1   1956    72060    510480   1304640   1382400    518400
  7 :  1  13699  1036224  12233340  50823360  94046400  79833600  25401600
  etc.

Cf. A000012 (column 0), A007526 (column 1), A001044 (main diagonal).

Cf. A000166, A048993, A131689, A320031, A371766, A371767.

T[n_, k_] := Sum[(-1)^(k - j)*Binomial[k, j]*HypergeometricPFQ[{1, -n}, {}, -j], {j, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten  (* Peter Luschny, Apr 12 2024 *)

T(n, k) = if(k==0, 1, if(k > n, 0, n*k*(T(n-1, k-1) + T(n-1, k))))

T(n, k) = Sum_{i=k..n} A131689(i, k) * n! / (n-i)!.
T(n, k) = n! * k! * (Sum_{i=0..n-k} A048993(n-i, k) / i!).
T(n, k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k, i) * A320031(n, i).
Conjecture: E.g.f. of column k is exp(t) * t^k * k! / (Prod_{i=0..k} (1 - i*t)).
Conjecture: Sum_{k=0..n} (-1)^(n-k) * T(n, k) = A000166(n).
T(n, k) = A371766(n, k) * A371767(n, k). - Peter Luschny, Apr 14 2024

A224869 a(n) = n*( a(n-1)+1 ), initialized by a(1) = -1.

Original entry on oeis.org

-1, 0, 3, 16, 85, 516, 3619, 28960, 260649, 2606500, 28671511, 344058144, 4472755885, 62618582404, 939278736075, 15028459777216, 255483816212689, 4598708691828420, 87375465144739999, 1747509302894800000, 36697695360790800021, 807349297937397600484

Offset: 1

Alex Ratushnyak, Jul 22 2013

			a(4) = 4*(a(3)+1) = 4*4 = 16.

Cf. A007526, A038156, A166554, A121662 (3rd column).

A224869[1] := -1; A224869[n_] := A224869[n - 1]n + n; Table[A224869[n], {n, 40}] (* Alonso del Arte, Jul 22 2013 *)

a=-1
for n in range(2,24):
  print(a, end=', ')
  a= n*(a+1)

a(n) +(-n-2)*a(n-1) +(2*n-1)*a(n-2) +(-n+2)*a(n-3)=0. - R. J. Mathar, Jul 28 2013

a(n) = exp(1)*n*Gamma(n,1) - 2*Gamma(n+1,0). - Martin Clever, Mar 27 2023

A318364 Expansion of e.g.f. exp(x*exp(x)/(1 - x)).

Original entry on oeis.org

1, 1, 5, 28, 197, 1676, 16597, 186796, 2350105, 32634928, 495207881, 8144456684, 144204493765, 2733218222944, 55188182951917, 1182163846918156, 26765995313355953, 638508459302742464, 16002492517241163793, 420279349847440766284, 11540406000681962458141, 330624627443307824367616

Offset: 0

Ilya Gutkovskiy, Aug 24 2018

nonn

Cf. A000248, A000262, A000522, A007526, A318365.

a:=series(exp(x*exp(x)/(1 - x)), x=0, 22): seq(n!*coeff(a, x, n), n=0..21); # Paolo P. Lava, Mar 26 2019

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

x = 'x + O('x^25); Vec(serlaplace(exp(x*exp(x)/(1 - x)))) \\ Michel Marcus, Aug 25 2018

a(0) = 1; a(n) = Sum_{k=1..n} A007526(k)*binomial(n-1,k-1)*a(n-k).

a(n) ~ exp(1/4 - 3*exp(1)/2 + 2*exp(1/2)*sqrt(n) - n) * n^(n - 1/4) / sqrt(2). - Vaclav Kotesovec, Aug 25 2018

A331797 E.g.f.: (exp(x) - 1) * exp(exp(x) - 1) / (2 - exp(x)).

Original entry on oeis.org

0, 1, 5, 28, 183, 1401, 12466, 127443, 1478581, 19239274, 277797577, 4409962349, 76355817104, 1432117088325, 28925947345561, 625973017346996, 14449435509751843, 354384392492622789, 9202836581079864186, 252260861877820739167, 7278710020682729662089

Offset: 0

Ilya Gutkovskiy, Jan 26 2020

nonn

Stirling transform of A007526.

Cf. A000110, A000670, A007526, A059099, A331796, A331798.

nmax = 20; CoefficientList[Series[(Exp[x] - 1) Exp[Exp[x] - 1]/(2 - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
A007526[n_] := n! Sum[1/k!, {k, 0, n - 1}]; a[n_] := Sum[StirlingS2[n, k] A007526[k], {k, 0, n}]; Table[a[n], {n, 0, 20}]
Table[(1/2) Sum[Binomial[n, k] HurwitzLerchPhi[1/2, -k, 0] BellB[n - k], {k, 1, n}], {n, 0, 20}]

a(n) = Sum_{k=0..n} Stirling2(n,k) * A007526(k).
a(n) = Sum_{k=1..n} binomial(n,k) * A000670(k) * A000110(n-k).
a(n) ~ n! * exp(1) / (2 * (log(2))^(n+1)). - Vaclav Kotesovec, Jan 26 2020

A334156 Triangle read by rows: T(n,m) is the number of length n decorated permutations avoiding the word 0^m = 0...0 of m 0's, where 1 <= m <= n.

Original entry on oeis.org

1, 2, 4, 6, 12, 15, 24, 48, 60, 64, 120, 240, 300, 320, 325, 720, 1440, 1800, 1920, 1950, 1956, 5040, 10080, 12600, 13440, 13650, 13692, 13699, 40320, 80640, 100800, 107520, 109200, 109536, 109592, 109600, 362880, 725760, 907200, 967680, 982800, 985824, 986328, 986400, 986409

Offset: 1

Jordan Weaver, Apr 16 2020

A length n decorated permutation is a word w = w_1....w_n on the letters {0,...,n} such that the restriction of w to its nonzero entries is an ordinary permutation in one-line notation. Then w avoids 0^m if w contains at most m-1 0's as letters, and w contains 0^m if w contains m 0's among its letters (not necessarily consecutive).

			For (n,m) = (3,2), the T(3,2) = 12 length 3 decorated permutations avoiding 0^2 = 00 are 012, 102, 120, 021, 201, 210, 123, 132, 213, 231, 312, and 321.
Triangle begins:
    1
    2,   4
    6,  12,  15
   24,  48,  60,  64
  120, 240, 300, 320, 325

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
S. Corteel, Crossings and alignments of permutations, Adv. Appl. Math 38 (2007) 149-163.
A. Postnikov, Total positivity, Grassmannians, and networks, arXiv:math/0609764 [math.CO], 2006.

Cf. A000142 (1st column), A007526 (right diagonal).

Row sums are A093964.

Array[Accumulate[#!/Range[0,#-1]!]&,10] (* Paolo Xausa, Jan 08 2024 *)

T(n,m)={sum(j=0, m-1, n!/j!)} \\ Andrew Howroyd, May 11 2020

T(n,m) = Sum_{j=0..m-1} n!/j!.

Terms a(37) and beyond from Andrew Howroyd, Jan 07 2024

cp's OEIS Frontend

A193563 a(0) = 0, a(n) = n^2 * (a(n-1) + 1).

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A285268 Triangle read by rows: T(m,n) = Sum_{i=1..n} P(m,i) where P(m,n) = m!/(m-n)! is the number of permutations of m items taken n at a time, for 1 <= n <= m.

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A357479 a(n) = (n!/6) * Sum_{k=0..n-3} 1/k!.

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A357480 a(n) = (n!/24) * Sum_{k=0..n-4} 1/k!.

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PARI

PARI

PARI

PARI

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A368576 a(n) = n! * Sum_{k=0..n} binomial(k+4,5) / k!.

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PARI

Formula

A371898 Triangle read by rows: T(n, k) = n * k * (T(n-1, k-1) + T(n-1, k)) for k > 0 with initial values T(n, 0) = 1 and T(i, j) = 0 for j > i.

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A224869 a(n) = n*( a(n-1)+1 ), initialized by a(1) = -1.

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A318364 Expansion of e.g.f. exp(x*exp(x)/(1 - x)).