A000522 -id:A000522 - cp's OEIS frontend

A080955 Square array of numbers related to the incomplete gamma function, read by antidiagonals.

1, 1, 1, 1, 2, 2, 1, 3, 5, 6, 1, 4, 10, 16, 24, 1, 5, 17, 38, 65, 120, 1, 6, 26, 78, 168, 326, 720, 1, 7, 37, 142, 393, 872, 1957, 5040, 1, 8, 50, 236, 824, 2208, 5296, 13700, 40320, 1, 9, 65, 366, 1569, 5144, 13977, 37200, 109601, 362880, 1, 10, 82, 538, 2760, 10970, 34960, 100026

Offset: 0

Paul Barry, Feb 26 2003

			Array begins:
k=0: 1 1 2 6 24 ...
k=1: 1 2 5 16 65 ...
k=2: 1 3 10 38 168 ...
k=3: 1 4 17 78 393 ...
k=4: 1 5 26 142 824 ...
...

Rows: A000142, A000522, A010842, A053486, A053487, A080954.

Transposed version: A089258.

T[0, k_] := k!; T[n_, k_] := k!*Sum[n^j/j!, {j, 0, k}];
Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 17 2018 *)

T(k,n) = n! * Sum{j=0..n} k^j/j!.
E.g.f. of k-th row: exp(k*x)/(1-x).
T(k,n) = A089258(n,k).

Corrected by Philippe Deléham, Dec 12 2003

A089258 Transposed version of A080955: T(n,k) = A080955(k,n), n>=0, k>=-1.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 5, 6, 9, 1, 4, 10, 16, 24, 44, 1, 5, 17, 38, 65, 120, 265, 1, 6, 26, 78, 168, 326, 720, 1854, 1, 7, 37, 142, 393, 872, 1957, 5040, 14833, 1, 8, 50, 236, 824, 2208, 5296, 13700, 40320, 133496, 1, 9, 65, 366, 1569, 5144, 13977, 37200, 109601, 362880, 1334961

Offset: 0

Philippe Deléham, Dec 12 2003

Can be extended to columns with negative indices k<0 via T(n,k) = A292977(n,-k). - Max Alekseyev, Mar 06 2018

			n\k -1   0   1    2    3    4     5     6  ...
----------------------------------------------
0  | 1,  1,  1,   1,   1,   1,    1,    1, ...
1  | 0,  1,  2,   3,   4,   5,    6,    7, ...
2  | 1,  2,  5,  10,  17,  26,   37,   50, ...
3  | 2,  6, 16,  38,  78, 152,  236,  366, ...
4  | 9, 24, 65, 168, 393, 824, 1569, 2760, ...
...

Columns: A000166, A000142, A000522, A010842, A053486, A053487, A080954.
Main diagonal gives A217701.
Cf. A080955, A008290, A292977.

(* Assuming offset (0, 0): *)
T[n_, k_] := Exp[k - 1] Gamma[n + 1, k - 1];
Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten  (* Peter Luschny, Dec 24 2021 *)

For n > 0, k >= -1, T(n,k) is the permanent of the n X n matrix with k+1 on the diagonal and 1 elsewhere.
T(0,k) = 1.
T(n,k) = Sum_{j>=0} A008290(n,j) * (k+1)^j.
T(n,k) = n*T(n-1, k) + k^n .
T(n,k) = n! * Sum_{j=0..n} k^j/j!.
E.g.f. for k-th column: exp(k*x)/(1-x).
Assuming n >= 0, k >= 0: T(n, k) = exp(k-1)*Gamma(n+1, k-1). - Peter Luschny, Dec 24 2021

Edited and changed offset for k to -1 by Max Alekseyev, Mar 08 2018

A093658 Lower triangular matrix, read by rows, defined as the convergent of the concatenation of matrices using the iteration: M(n+1) = [[M(n),0*M(n)],[M(n)^2,M(n)]], with M(0) = [1].

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 1, 1, 2, 0, 1, 0, 1, 0, 1, 6, 2, 2, 1, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 1, 1, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 6, 2, 2, 1, 0, 0, 0, 0, 2, 1, 1, 1, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 6, 2, 0, 0, 2, 1, 0, 0, 2, 1, 0, 0, 1, 1

Offset: 1

Paul D. Hanna, Apr 08 2004

Related to factorials, the incomplete gamma function (A010842) and the total number of arrangements of sets (A000522).
First column forms A093659, where A093659(2^n) = n! for n>=0.
Row sums form A093660, where A093660(2^n) = A000522(n) for n>=0.
Partial sums of the row sums form A093661, where A093661(2^n) = A010842(n) for n>=0.

			Let M(n) be the lower triangular matrix formed from the first 2^n rows.
To generate M(3) from M(2), take the matrix square of M(2):
[1,0,0,0]^2=[1,0,0,0]
[1,1,0,0]...[2,1,0,0]
[1,0,1,0]...[2,0,1,0]
[2,1,1,1]...[6,2,2,1]
and append M(2)^2 to the bottom left corner and M(2) to the bottom right:
[1],
[1,1],
[1,0,1],
[2,1,1,1],
.........
[1,0,0,0],[1],
[2,1,0,0],[1,1],
[2,0,1,0],[1,0,1],
[6,2,2,1],[2,1,1,1].
Repeating this process converges to triangle A093658.

Cf. A000522, A010842, A093655, A093662.

T(2^n, 1) = n! for n>=0.

A123899 a(n) = (n+1)!/(d(n)*d(n+1)) where d(n) = cancellation factor in reducing Sum_{k=0...n} 1/k! to lowest terms.

Original entry on oeis.org

1, 2, 3, 12, 60, 360, 252, 2016, 36288, 362880, 4989600, 11975040, 622702080, 8717829120, 65383718400, 5230697472000, 2736057139200, 49249028505600, 30411275102208, 608225502044160, 25545471085854720000

Offset: 0

Jonathan Sondow, Oct 18 2006

			a(2) = 3 because (2+1)!/(d(2)*d(3)) = 3!/(gcd(2,5)*gcd(6,16)) = 6/2 = 3.

J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, arXiv:0704.1282 [math.HO], 2007-2010.
J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.

Cf. A000522, A061354, A093101, A123900, A123901.

(A[n_] := If[n==0,1,n*A[n-1]+1]; d[n_] := GCD[A[n],n! ]; Table[(n+1)!/(d[n]*d[n+1]), {n,0,22}])

a(n) = (n+1)!/(A093101(n)*A093101(n+1)) where A093101(n)=gcd(n!,1+n+n(n-1)+...+n!).

A123900 a(n) = (n+3)!/(d(n)d(n+1)d(n+2)) where d(n) = cancellation factor in reducing Sum_{k=0...n} 1/k! to lowest terms.

Original entry on oeis.org

6, 12, 60, 180, 2520, 1008, 18144, 18144, 3991680, 5987520, 155675520, 1089728640, 26153487360, 523069747200, 17784371404800, 12312257126400, 935731541606400, 4678657708032, 12772735542927360, 140500090972200960

Offset: 0

Jonathan Sondow, Oct 18 2006

			a(2) = 60 because (2+3)!/(d(2)*d(3)*d(4)) = 5!/(GCD(2,5)*GCD(6,16)*GCD(24,65)) = 120/2 = 60.

J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, arXiv:0704.1282 [math.HO], 2007-2010.
J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.

Cf. A000522, A061354, A093101, A123899, A123901.

(A[n_] := If[n==0,1,n*A[n-1]+1]; d[n_] := GCD[A[n],n! ]; Table[(n+3)!/(d[n]*d[n+1]*d[n+2]), {n,0,21}])

a(n) = (n+3)!/(A093101(n)*A093101(n+1)*A093101(n+2)) where A093101(n) = gcd(n!,1+n+n(n-1)+...+n!).

A286286 a(0) = 0; thereafter, a(n) = (2n-1)a(n-1) + 1.

Original entry on oeis.org

0, 1, 4, 21, 148, 1333, 14664, 190633, 2859496, 48611433, 923617228, 19395961789, 446107121148, 11152678028701, 301122306774928, 8732546896472913, 270708953790660304, 8933395475091790033, 312668841628212651156, 11568747140243868092773

Offset: 0

N. J. A. Sloane, May 15 2017

Seiichi Manyama, Table of n, a(n) for n = 0..404

Conjectured to give indices of records in A132424.

Cf. A001147, A002627 (similar sequence), A000522, A060196.

NestList[{(2 #2 - 1) #1 + 1, #2 + 1} & @@ # &, {0, 1}, 19][[All, 1]] (* Michael De Vlieger, Dec 10 2021 *)

a(n) = (2*n-1)!! * Sum_{k=1..n} 1/(2*k-1)!!. - Seiichi Manyama, Sep 02 2017
a(n) = floor((2*n-1)!!*A060196), for n > 0. - Peter McNair, Dec 10 2021
From Peter Bala, Feb 09 2024: (Start)
a(n) = 2*n*a(n-1) - (2*n - 3)*a(n-2) with a(0) = 0 and a(1) = 1.
The double factorial numbers (2*n-1)!! = A001147(n) satisfy the same recurrence, leading to the generalized continued fraction expansion Limit_{n -> oo} a(n)/(2*n-1)!! = Sum_{k >= 1} 1/(2*k-1)!! = A060196 = 1/(1 - 1/(4 - 3/(6 - 5/(8 - 7/(10 - 9/(12 - ... )))))). (End)

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

A371318 E.g.f. satisfies A(x) = exp(x) + x*A(x)^3.

Original entry on oeis.org

1, 2, 13, 190, 4345, 135346, 5345749, 256004974, 14416470961, 933597699202, 68358972056221, 5584583237569150, 503607231488672425, 49690178089937051122, 5325031693664693833957, 615922452708451717999726, 76479190243720703567763553

Offset: 0

Seiichi Manyama, Mar 18 2024

nonn

Cf. A000522, A194471, A373324.

Table[n! Sum[(2 k + 1)^(n - k - 1)*Binomial[3 k, k]/(n - k)!, {k, 0, n}], {n, 0, 20}] (* Wesley Ivan Hurt, May 25 2024 *)

a(n) = n!*sum(k=0, n, (2*k+1)^(n-k-1)*binomial(3*k, k)/(n-k)!);

a(n) = n! * Sum_{k=0..n} (2*k+1)^(n-k-1) * binomial(3*k,k)/(n-k)!.

a(n) ~ sqrt(1 + LambertW(8/27)) * 2^n * n^(n-1) / (3 * exp(n) * LambertW(8/27)^(n + 1/2)). - Vaclav Kotesovec, Jun 01 2024

A056547 a(n) = 6na(n-1) + 1 with a(0)=1.

Original entry on oeis.org

1, 7, 85, 1531, 36745, 1102351, 39684637, 1666754755, 80004228241, 4320228325015, 259213699500901, 17108104167059467, 1231783500028281625, 96079113002205966751, 8070645492185301207085, 726358094296677108637651

Offset: 0

Henry Bottomley, Jun 20 2000

			a(2) = 6*2*a(1) + 1 = 12*7 + 1 = 85.

Harvey P. Dale, Table of n, a(n) for n = 0..346

Cf. A000522, A010844, A010845, A056545, A056546 for analogs. A056547/(A000142*A000400) is an increasingly good approximation to 6th root of e.

nxt[{n_,a_}]:={n+1,6a(n+1)+1}; NestList[nxt,{0,1},20][[;;,2]] (* Harvey P. Dale, Jul 17 2024 *)

a(n) = floor(e^(1/6)*6^n*n!).
a(n) = n!*Sum_{k=0..n} 6^(n-k)/k!. E.g.f.: exp(x)/(1 - 6*x). - Philippe Deléham, Mar 14 2004
From Peter Bala, Mar 01 2017: (Start)
a(n) = Integral_{x = 0..inf} (6*x + 1)^n*exp(-x) dx.
The e.g.f. y = exp(x)/(1 - 6*x) satisfies the differential equation (1 - 6*x)*y' = (7 - 6*x)*y.
a(n) = (6*n + 1)*a(n-1) - 6*(n - 1)*a(n-2).
The sequence b(n) := 6^n*n! also satisfies the same recurrence with b(0) = 1, b(1) = 6. This leads to the continued fraction representation a(n) = 6^n*n!*( 1 + 1/(6 - 6/(13 - 12/(19 - ... - (6*n - 6)/(6*n + 1) )))) for n >= 2. Taking the limit gives the continued fraction representation exp(1/6) = 1 + 1/(6 - 6/(13 - 12/(19 - ... - (6*n - 6)/((6*n + 1) - ... )))). Cf. A010844. (End)

More terms from James Sellers, Jul 04 2000

A076571 Binomial triangle based on factorials.

Original entry on oeis.org

1, 1, 2, 2, 3, 5, 6, 8, 11, 16, 24, 30, 38, 49, 65, 120, 144, 174, 212, 261, 326, 720, 840, 984, 1158, 1370, 1631, 1957, 5040, 5760, 6600, 7584, 8742, 10112, 11743, 13700, 40320, 45360, 51120, 57720, 65304, 74046, 84158, 95901, 109601

Offset: 0

Henry Bottomley, Oct 19 2002

			Rows start:
    1;
    1,   2;
    2,   3,   5;
    6,   8,  11,  16;
   24,  30,  38,  49,  65;
  120, 144, 174, 212, 261, 326;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
E. Biondi, L. Divieti, and G. Guardabassi, Counting paths, circuits, chains and cycles in graphs: A unified approach, Canad. J. Math. 22 1970 22-35. See Table I.
D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.

Columns include A000142, A001048, A001344, A001345, A001346, A001347.
Right hand columns include A000522, A001339, A001340, A001341, A001342.
Cf. A002627 (row sums), A099022.

A076571:= func< n,k| (&+[Binomial(k,j)*Factorial(n-j): j in [0..k]]) >;
[A076571(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 05 2023

A076571[n_, k_]:= n!*Hypergeometric1F1[-k,-n,1];
Table[A076571[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 05 2023 *)

def A076571(n,k): return sum(binomial(k,j)*factorial(n-j) for j in range(k+1))
flatten([[A076571(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 05 2023

T(n, k) = Sum_{j=0..k} binomial(k, j)*(n-j)!.
T(n, k) = T(n, k-1) + T(n-1, k-1) with T(n, 0) = n!.
T(n, n) = A000522(n).
Sum_{k=0..n} T(n, k) = A002627(n+1).
From G. C. Greubel, Oct 05 2023: (Start)
T(n, k) = n! * Hypergeometric1F1([-k], [-n], 1).
T(2*n, n) = A099022(n). (End)

cp's OEIS Frontend

A080955 Square array of numbers related to the incomplete gamma function, read by antidiagonals.

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A089258 Transposed version of A080955: T(n,k) = A080955(k,n), n>=0, k>=-1.

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A093658 Lower triangular matrix, read by rows, defined as the convergent of the concatenation of matrices using the iteration: M(n+1) = [[M(n),0*M(n)],[M(n)^2,M(n)]], with M(0) = [1].

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A123899 a(n) = (n+1)!/(d(n)*d(n+1)) where d(n) = cancellation factor in reducing Sum_{k=0...n} 1/k! to lowest terms.

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A123900 a(n) = (n+3)!/(d(n)*d(n+1)*d(n+2)) where d(n) = cancellation factor in reducing Sum_{k=0...n} 1/k! to lowest terms.

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A286286 a(0) = 0; thereafter, a(n) = (2*n-1)*a(n-1) + 1.

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

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A371318 E.g.f. satisfies A(x) = exp(x) + x*A(x)^3.

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A056547 a(n) = 6*n*a(n-1) + 1 with a(0)=1.

A123900 a(n) = (n+3)!/(d(n)d(n+1)d(n+2)) where d(n) = cancellation factor in reducing Sum_{k=0...n} 1/k! to lowest terms.

A286286 a(0) = 0; thereafter, a(n) = (2n-1)a(n-1) + 1.

A056547 a(n) = 6na(n-1) + 1 with a(0)=1.