A010842 -id:A010842 - cp's OEIS frontend

A093661 Partial sums of A093660.

1, 3, 5, 10, 12, 17, 22, 38, 40, 45, 50, 66, 71, 87, 103, 168, 170, 175, 180, 196, 201, 217, 233, 298, 303, 319, 335, 400, 416, 481, 546, 872, 874, 879, 884, 900, 905, 921, 937, 1002, 1007, 1023, 1039, 1104, 1120, 1185, 1250, 1576, 1581, 1597, 1613, 1678, 1694

Offset: 1

Paul D. Hanna, Apr 08 2004

nonn

A093660 gives the row sums of lower triangular matrix A093658. Related to the incomplete Gamma Function at 2 (A010842).

Cf. A010842, A093658, A093660.

a(2^n) = A010842(n) for n>=0.

A108869 E.g.f. : exp(6x)/(1-x).

Original entry on oeis.org

1, 7, 50, 366, 2760, 21576, 176112, 1512720, 13781376, 134110080, 1401566976, 15780033792, 191537187840, 2503044135936, 35120982067200, 527284915992576, 8439379765788672, 143486382677852160, 2582856448158007296

Offset: 0

Philippe Deléham, Jul 13 2005

Binomial transform of A080954.

a(n) is the permanent of the n X n matrix with 7's on the diagonal and 1's elsewhere.

Amiram Eldar, Table of n, a(n) for n = 0..448

Cf. A000142, A000522, A008290, A010842, A053486, A053487, A080954.

a:=n->n!*sum(6^k/k!,k=0..n): seq(a(n),n=0..20); # Emeric Deutsch, Jul 18 2005
restart:F(x):=exp(6*x)/(1-x): f[0]:=F(x): for n from 1 to 20 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..18); # Zerinvary Lajos, Apr 03 2009

a[n_] := n! * Sum[6^k/k!, {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Jun 30 2020 *)

a(n) = n!*Sum_{ k = 0..n } 6^k/k!.
a(n) = Sum_{ k = 0..n } A008290(n, k)*7^k.
a(n) Sum_{ k = 0..n } k!*C(n, k)*6^(n-k).

More terms from Emeric Deutsch, Jul 18 2005

A343685 a(0) = 1; a(n) = 2 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).

Original entry on oeis.org

1, 3, 19, 182, 2328, 37234, 714674, 16004064, 409587144, 11792756640, 377261048592, 13275818803488, 509646721402032, 21195285059025648, 949279217570464944, 45552467588773815744, 2331624264279599225088, 126804353256754734370176, 7301857349340031590836352, 443826900013575494233057536

Offset: 0

Ilya Gutkovskiy, Apr 26 2021

nonn

Cf. A007840, A010842, A052820, A343686, A343687.

a[0] = 1; a[n_] := a[n] = 2 n a[n - 1] + Sum[Binomial[n, k] (n - k - 1)! a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 19}]
nmax = 19; CoefficientList[Series[1/(1 - 2 x + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: 1 / (1 - 2*x + log(1 - x)).

a(n) ~ n! / ((2/c + 1 - c) * (1 - c/2)^n), where c = LambertW(2*exp(1)) = 1.3748225281836233816178373171119... - Vaclav Kotesovec, Apr 26 2021

A080252 a(n) = na(n-1)+4a(n-2)-4(n-2)a(n-3).

Original entry on oeis.org

0, 1, 2, 10, 40, 216, 1296, 9136, 73088, 658048, 6580480, 72386304, 868635648, 11292267520, 158091745280, 2371376195584, 37942019129344, 645014325264384, 11610257854758912, 220594899240681472, 4411897984813629440

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Feb 10 2003

Cf. A000023, A010842.

A080252 := n -> (exp(2)*GAMMA(1+n,2) - exp(-2)*GAMMA(1+n,-2))/4:
seq(simplify(A080252(n)), n=0..20); # Peter Luschny, Dec 18 2017

c = CoefficientList[Series[(Sinh[z]*Cosh[z])/(1 - z), {z, 0, 25}], z]; For[n = 0, n < 25, n++; Print[c[[n]]*(n - 1)! ]]
RecurrenceTable[{a[0]==0,a[1]==1,a[2]==2,a[n]==n*a[n-1]+4a[n-2]-4(n-2)* a[n-3]}, a,{n,20}] (* Harvey P. Dale, Nov 17 2013 *)

x='x+O('x^99); concat([0], Vec(serlaplace(sinh(x)*cosh(x)/(1-x)))) \\ Altug Alkan, Dec 18 2017

E.g.f.: sinh(z)*cosh(z)/(1-z).
a(n) ~ n!*(e^2-1/e^2)/4. - Vaclav Kotesovec, Oct 13 2012
a(n) = (exp(2)*Gamma(1+n,2) - exp(-2)*Gamma(1+n,-2))/4 = (A010842(n) - A000023(n))/4. - Peter Luschny, Dec 18 2017

A188143 Binomial transform of A187848.

Original entry on oeis.org

1, 5, 29, 193, 1453, 12209, 113237, 1149241, 12675661, 151095569, 1937411429, 26614052617, 390244490749, 6087782363009, 100728768290645, 1762767028074937, 32542231109506285, 632202858036492593, 12895661952702667205

Offset: 0

Groux Roland, Mar 24 2011

nonn

a(n) is also the INVERTi transform of A010842(n+1) starting at n=2.
a(n) is also the moment of order n for the measure of density: exp(x-2) / ((Ei(x-2))^2+Pi^2) over the interval 2..infinity with Ei the exponential integral.
More generally, for every integer k, the sequence a(n,k)=int(x^n*exp(x-k) / ((Ei(x-k))^2+Pi^2), x=k..infinity) is the INVERTi transform of the sequence b(n+1,k), starting at n=2, with b(n,k)=int(x^n*exp(x-k), x=k..infinity) whose e.g.f. is exp(k*x)/(1-x).

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A000023.

with(LinearAlgebra):
c:= proc(n) option remember; add(n!/k!, k=0..n) end:
b:= n-> (-1)^(n+1) * Determinant(Matrix(n+2,
        (i, j)-> `if`(0<=i+1-j, c(i+1-j), 0))):
a:= proc(n) add(b(k) *binomial(n,k), k=0..n) end:
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 24 2011

c[n_] := c[n] = Sum[n!/k!, {k, 0, n}]; b[n_] := (-1)^(n+1)*Det[Table[If[0 <= i+1-j, c[i+1-j], 0], {i, 1, n+2}, {j, 1, n+2}]]; a[n_] := Sum[b[k] * Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 28 2017, after Alois P. Heinz *)

a(n) = Integral_{x=2..oo} x^n*exp(x-2)/((Ei(x-2))^2 + Pi^2) dx.
G.f.: 1/x^2 - 3/x - Q(0)/x^2, where Q(k) = 1 - 2*x - x*(k+1)/(1-x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 19 2013
a(n) ~ exp(2) * n^2 * n!. - Vaclav Kotesovec, Nov 02 2023

A295518 a(n) = e^2 * Sum_{k=0..n-1} Gamma(k + 1, 2).

Original entry on oeis.org

0, 1, 4, 14, 52, 220, 1092, 6388, 43588, 341444, 3022660, 29835844, 324782916, 3864151876, 49875956548, 694041238340, 10356520497988, 164956188717892, 2793150548587332, 50100649026499396, 948943120107352900, 18925792541725471556, 396439630395708060484

Offset: 0

Peter Luschny, Dec 17 2017

nonn

Cf. A010842.

a := proc(n) option remember; if n=0 then 0 elif n=1 then 1 elif n=2 then 4 else
(2*n-4)*a(n-3)+(3-3*n)*a(n-2)+(2+n)*a(n-1) fi end: seq(a(n), n=0..22);

a[n_] := E^2 Sum[Gamma[k + 1, 2], {k,0,n-1}]; Table[a[n], {n,0,22}]

a(n) = (2*n-4)*a(n-3)+(3-3*n)*a(n-2)+(2+n)*a(n-1) for n >= 3.

A336997 a(n) = n! * Sum_{d|n} 2^(d - 1) / d!.

Original entry on oeis.org

1, 4, 10, 56, 136, 1952, 5104, 94208, 605056, 7741952, 39917824, 1458295808, 6227024896, 175463616512, 2353813878784, 48886264659968, 355687428161536, 17362063156969472, 121645100409094144, 6001501553433509888, 85800344155030552576, 2248030289949388439552

Offset: 1

Ilya Gutkovskiy, Aug 10 2020

nonn

Cf. A010842, A034729, A057625, A336998.

Table[n! Sum[2^(d - 1)/d!, {d, Divisors[n]}], {n, 1, 22}]
nmax = 22; CoefficientList[Series[Sum[(Exp[2 x^k] - 1)/2, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest

a(n) = n! * sumdiv(n, d, 2^(d-1)/d!); \\ Michel Marcus, Aug 12 2020

E.g.f.: Sum_{k>=1} (exp(2*x^k) - 1) / 2.

a(p) = p! + 2^(p - 1), where p is prime.

A134558 Array read by antidiagonals, a(n,k) = gamma(n+1,k)*e^k, where gamma(n,k) is the upper incomplete gamma function and e is the exponential constant 2.71828...

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 5, 3, 1, 24, 16, 10, 4, 1, 120, 65, 38, 17, 5, 1, 720, 326, 168, 78, 26, 6, 1, 5040, 1957, 872, 393, 142, 37, 7, 1, 40320, 13700, 5296, 2208, 824, 236, 50, 8, 1, 362880, 109601, 37200, 13977, 5144, 1569, 366, 65, 9, 1, 3628800, 986410, 297856

Offset: 0

Ross La Haye, Jan 22 2008

			Square array begins:
    1,    1,    1,     1,     1,     1,      1, ...
    1,    2,    3,     4,     5,     6,      7, ...
    2,    5,   10,    17,    26,    37,     50, ...
    6,   16,   38,    78,   142,   236,    366, ...
   24,   65,  168,   393,   824,  1569,   2760, ...
  120,  326,  872,  2208,  5144, 10970,  21576, ...
  720, 1957, 5296, 13977, 34960, 81445, 176112, ...

Eric Weisstein's World of Mathematics, Incomplete Gamma Function.
Wikipedia, Incomplete gamma function.

Cf. a(n, 0) = A000142(n); a(n, 1) = A000522(n); a(n, 2) = A010842(n); a(n, 3) = A053486(n); a(n, 4) = A053487(n); a(n, 5) = A080954(n); a(n, 6) = A108869(n); a(1, k) = A000027(k+1); a(2, k) = A002522(k+1); a(n, n) = A063170(n); a(n, n+1) = A001865(n+1); a(n, n+2) = A001863(n+2).
Another version: A089258.
A transposed version: A080955.
Cf. A001113.

T[n_,k_] := Gamma[n+1, k]*E^k; Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] //Flatten (* Amiram Eldar, Jun 27 2020 *)

a(n,k) = gamma(n+1,k)*e^k = Sum_{m=0..n} m!*binomial(n,m)*k^(n-m).
a(n,k) = n*a(n-1,k) + k^n for n,k > 0.
E.g.f. (by columns) is e^(kx)/(1-x).
a(n,k) = the binomial transform by columns of a(n,k-1).
Conjecture: a(n,k) is the permanent of the n X n matrix with k+1 on the main diagonal and 1 elsewhere.

More terms from Amiram Eldar, Jun 27 2020

A180435 a(n) = a(n-1)*2^n+n, a(0)=1.

Original entry on oeis.org

1, 3, 14, 115, 1844, 59013, 3776838, 483435271, 123759429384, 63364827844617, 64885583712887818, 132885675443994251275, 544299726618600453222412, 4458903360459574912797999117, 73054672657769675371282417532942

Offset: 0

Georgi Guninski, Sep 05 2010

nonn

Cf. A010842.

nxt[{n_,a_}]:={n+1,a*2^(n+1)+n+1}; Transpose[NestList[nxt,{0,1},20]] [[2]] (* Harvey P. Dale, Apr 05 2015 *)

PARI
```
a(n)=if(n<=0,1, a(n-1)*2^n+n )
```

a(n+1) = (2^(n + 1) + 1)*a(n) - 2^n*a(n - 1) + 1.
a(n+1) = ((a(n - 2) + 4*a(n - 1) + 4)*a(n) - 2*a(n - 1)^2 - 4*a(n)^2 + a(n - 2) - 4*a(n - 1))/(a(n - 2) - 2*a(n - 1)).
a(n) = 2^(n*(n+1)/2) + sum_{k=1..n} 2^( (n+k+1)*(n-k)/2 ) * k. - Max Alekseyev, Sep 05 2010

Minor edits by N. J. A. Sloane, Sep 05 2010

A371686 Triangle read by rows: T(n, k) = e * binomial(n, k) * Gamma(k + 1, 1).

Original entry on oeis.org

1, 1, 2, 1, 4, 5, 1, 6, 15, 16, 1, 8, 30, 64, 65, 1, 10, 50, 160, 325, 326, 1, 12, 75, 320, 975, 1956, 1957, 1, 14, 105, 560, 2275, 6846, 13699, 13700, 1, 16, 140, 896, 4550, 18256, 54796, 109600, 109601, 1, 18, 180, 1344, 8190, 41076, 164388, 493200, 986409, 986410

Offset: 0

Peter Luschny, Apr 03 2024

			Triangle starts:
  [0] 1;
  [1] 1,  2;
  [2] 1,  4,   5;
  [3] 1,  6,  15,  16;
  [4] 1,  8,  30,  64,   65;
  [5] 1, 10,  50, 160,  325,  326;
  [6] 1, 12,  75, 320,  975, 1956,  1957;
  [7] 1, 14, 105, 560, 2275, 6846, 13699, 13700;

Cf. A000522 (main diagonal), A007526 (subdiagonal), A010842 (row sums), A000142 and A133942 (alternating row sums), A367963 (central terms).

T := (n, k) -> binomial(n, k)*GAMMA(k + 1, 1)*exp(1):
seq(seq(simplify(T(n, k)), k = 0..n), n = 0..9);

T[n_,k_]:=(n!/(n-k)!)*Sum[1/j!,{j,0,k}];Flatten[Table[T[n,k],{n,0,9},{k,0,n}]] (* Detlef Meya, Apr 06 2024 *)

T(n, k) = (n! / (n - k)!)*(Sum_{j = 0..k} (1 / j!)). - Detlef Meya, Apr 06 2024

cp's OEIS Frontend

A093661 Partial sums of A093660.

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A108869 E.g.f. : exp(6x)/(1-x).

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A343685 a(0) = 1; a(n) = 2 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).

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A080252 a(n) = n*a(n-1)+4*a(n-2)-4*(n-2)*a(n-3).

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A188143 Binomial transform of A187848.

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A295518 a(n) = e^2 * Sum_{k=0..n-1} Gamma(k + 1, 2).

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A336997 a(n) = n! * Sum_{d|n} 2^(d - 1) / d!.

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A134558 Array read by antidiagonals, a(n,k) = gamma(n+1,k)*e^k, where gamma(n,k) is the upper incomplete gamma function and e is the exponential constant 2.71828...

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A180435 a(n) = a(n-1)*2^n+n, a(0)=1.

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A080252 a(n) = na(n-1)+4a(n-2)-4(n-2)a(n-3).