A001339 -id:A001339 - cp's OEIS frontend

A076571 Binomial triangle based on factorials.

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)

A081923 Expansion of e.g.f.: exp(2x)/(1-x)^2.

Original entry on oeis.org

1, 4, 18, 92, 536, 3552, 26608, 223456, 2085504, 21450752, 241320704, 2949474816, 38933066752, 552141672448, 8374148696064, 135274709700608, 2318995023429632, 42051109758173184, 804227474125029376

Offset: 0

Paul Barry, Apr 01 2003

Binomial transform of A001339.
Polynomials in A010027 evaluated at 3. - Ralf Stephan, Dec 15 2004
From Dennis P. Walsh, Sep 18 2013: (Start)
a(n) is the number of rooted labeled forests that satisfy the following conditions:
(i) there are 4 roots labeled 1, 2, 3, and 4;
(ii) there are n non-root vertices labeled 5,..., n+4;
(iii) the trees with roots 1 and 2 have width one;
(iv) the trees with roots 3 and 4 have height at most one.
To construct such a forest, for k=0,...,n, we take the following steps:
(1) choose k non-root vertices for trees with roots 1 and 2;
(2) construct width-one trees on roots 1 and 2 with the k non-root vertices;
(3) with the n-k remaining non-root vertices construct trees of height at most one on roots 3 and 4.
Thus a(n) is the sum (over k) of the product of the number of ways to do each step: a(n)=sum(k=0..n, binomial(n,k)*(k+1)!*2^(n-k)).  (End)

			For n=2, the a(2)=18 forests that satisfy the specified conditions are given in the link above. - _Dennis P. Walsh_, Sep 20 2013

Harvey P. Dale, Table of n, a(n) for n = 0..448
Paul Barry, A note on number triangles that are almost their own production matrix, arXiv:1804.06801 [math.CO], 2018.
Dennis P. Walsh, 18 Forests

Cf. A081924. A010842.

Cf. A081923(n) (sum(k=0..n, binomial(n,k)*A000522(n-k)*A000522(k))).

seq(n!*add((k+1)*2^(n-k)/(n-k)!,k=0..n),n=0..40); # Dennis P. Walsh, Sep 18 2013
seq(simplify(KummerU(-n, -n - 1, 2)), n = 0..24); # Peter Luschny, May 10 2022

With[{nn=20},CoefficientList[Series[Exp[2x]/(1-x)^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 10 2025 *)

E.g.f.: exp(2*x)/(1-x)^2
E.g.f.: 1/U(0) where U(k)=   1 - 2*x/( 1 + x/(2 - x - 4/( 2 - x*(k+1)/U(k+1)))) ; (continued fraction, 3rd kind, 4-step). - Sergei N. Gladkovskii, Oct 28 2012
Conjecture: a(n) +(-n-3)*a(n-1) +2*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 24 2012
G.f.: 2/x/G(0) - 1/x, where G(k)= 1 + 1/(1 - x*(2*k+2)/(x*(2*k+4) - 1 + x*(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 31 2013
G.f.: (sum(k>=0, k!*(x/(1-2*x))^k ) - 1)/x = Q(0)/(2*x) - 1/x, where Q(k)= 1 + 1/(1 - x*(k+1)/(x*(k+1) + (1-2*x)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 09 2013
G.f.: W(0)/x - 1/x, where W(k) = 1 - x*(k+1)/( x*(k+3) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013
a(n) = n!*sum(k=0..n, (k+1)*2^(n-k)/(n-k)!). [Dennis P. Walsh, Sep 18 2013]
a(n) = n!*sum(k=0..n, (n-k+1)*2^k/k!). [Dennis P. Walsh, Sep 18 2013]
From Peter Bala, Sep 25 2013: (Start)
a(n) ~ n!*n*e^2.
Applying Maple's ZeilbergerRecurrence command to the above series of Walsh for a(n) results in the first-order recurrence equation (n - 1)*a(n+1) = n*(n + 1)*a(n) - 2^(n+2) with a(0) = 1 and a(2) = 18. Using this it is easy to verify that a(n) satisfies the second-order recurrence a(n) = (n + 3)*a(n-1) - 2*(n - 1)*a(n-2) conjectured above by Mathar.
The sequence b(n) = n!*(n - 1) satisfies the same second-order recurrence but with the initial conditions b(0) = -1 and b(1) = 0. This leads to the finite continued fraction expansion a(n)/b(n) = 9 - 2*( 4/(6 - 6/(7 - 8/(9 - ... - 2*n/(n + 4)))) ) valid for n >= 2. Letting n tend to infinity produces the infinite continued fraction expansion e^2 = 9 - 2*( 4/(6 - 6/(7 - 8/(9 - ... - 2*n/(n + 4 - ...)))) ). (End)
a(n) = KummerU(-n, -n - 1, 2). - Peter Luschny, May 10 2022

Definition clarified by Harvey P. Dale, May 10 2025

A082427 a(1)=1, a(n) = n * (Sum_{k=1..n-1} a(k)) - 2.

Original entry on oeis.org

1, 0, 1, 6, 38, 274, 2238, 20462, 207178, 2301978, 27853934, 364633318, 5135252562, 77423807858, 1244311197718, 21236244441054, 383579665216538, 7310577148832842, 146617686151591998, 3086688129507199958, 68061473255633759074, 1568654907415559018658

Offset: 1

Benoit Cloitre, Apr 24 2003

nonn

Cf. A007808, A074143, A082425, A082428, A082430, A383436, A383437.

Cf. A001339.

Join[{1},Table[Floor[n(11/2-2E)n!],{n,2,20}]] (* Harvey P. Dale, May 09 2013 *)

a(n) = floor(n*(11/2 - 2*e)*n!) for n >= 2.
a(n) = (n+2)*a(n-1) - (n-1)*a(n-2) for n>3. - Gary Detlefs, Jun 30 2024
From Seiichi Manyama, Apr 27 2025: (Start)
E.g.f.: 2 + 3*x/2 + (11*x/2 - 2*exp(x))/(1-x)^2.
a(n) = 11*n/2 * n! - 2 * Sum_{k=0..n} (k+1)! * binomial(n,k) for n > 1.
a(n) = (n^2 * a(n-1) + 2)/(n-1) for n > 2. (End)

Offset changed to 1 by Georg Fischer, May 15 2024

A082428 a(1) = 1; a(n) = 3 + n * Sum_{k=1..n-1} a(k).

Original entry on oeis.org

1, 5, 21, 111, 693, 4989, 40743, 372507, 3771633, 41907033, 507075099, 6638074023, 93486209157, 1409484384213, 22652427603423, 386601431098419, 6982988349215193, 133087542655630737, 2669144605482372003, 56192518010155200063, 1239045022123922161389, 28557037652760872672013

Offset: 1

Benoit Cloitre, Apr 24 2003

nonn

Cf. A007808, A074143, A082425, A082427, A082430, A383436, A383437.

Cf. A001339.

for n>=2 a(n) = ceiling(n*(3e-7)*n!).
From Seiichi Manyama, Apr 27 2025: (Start)
E.g.f.: -3 - x + (-7*x + 3*exp(x))/(1-x)^2.
a(n) = -7 * n * n! + 3 * Sum_{k=0..n} (k+1)! * binomial(n,k) for n > 1.
a(n) = (n^2 * a(n-1) - 3)/(n-1) for n > 2.
a(n) = (n+2) * a(n-1) - (n-1) * a(n-2) for n > 3. (End)

A269951 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)C(-j,-n)S1(j,k), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 5, 5, 1, 0, 16, 23, 9, 1, 0, 65, 116, 65, 14, 1, 0, 326, 669, 470, 145, 20, 1, 0, 1957, 4429, 3634, 1415, 280, 27, 1, 0, 13700, 33375, 30681, 14084, 3535, 490, 35, 1, 0, 109601, 283072, 284066, 147532, 43939, 7756, 798, 44, 1

Offset: 0

Peter Luschny, Apr 10 2016

			Triangle begins:
  1;
  0,   1;
  0,   2,   1;
  0,   5,   5,   1;
  0,  16,  23,   9,   1;
  0,  65, 116,  65,  14,  1;
  0, 326, 669, 470, 145, 20, 1;

Peter Luschny, Extensions of the binomial

A000522 (col. 1), A073596 (col. 2), A000096 (diag. n-1), A241765 (diag. n-2).
A001339 (row sums), A137597 (unsigned matrix inverse).
Cf. A132393, A269952.

A269951 := (n,k) -> add((-1)^(n-j)*binomial(-j,-n)*abs(Stirling1(j,k)), j=0..n):
seq(seq(A269951(n,k), k=0..n), n=0..9);

Flatten[ Table[ Sum[(-1)^(n-j) Binomial[-j,-n] Abs[StirlingS1[j,k]], {j,0,n}], {n,0,9}, {k,0,n}]]

A358738 Expansion of Sum_{k>=0} k! * ( x/(1 - k*x) )^k.

Original entry on oeis.org

1, 1, 3, 15, 103, 893, 9341, 114355, 1603155, 25318137, 444689497, 8597568671, 181430298479, 4149361409077, 102229328244837, 2699254206069387, 76038064580742091, 2276259442660623857, 72160287650141753777, 2414950992007231422007

Offset: 0

Seiichi Manyama, Nov 29 2022

nonn

Cf. A001339, A006153, A080108.

Cf. A358740, A358741.

nmax = 20; CoefficientList[Series[Sum[k! * (x/(1 - k*x))^k, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 18 2023 *)

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, k!*(x/(1-k*x))^k))

a(n) = if(n==0, 1, sum(k=1, n, k!*k^(n-k)*binomial(n-1, k-1)));

a(n) = Sum_{k=1..n} k! * k^(n-k) * binomial(n-1,k-1) for n > 0.

a(n) ~ n! / ((1 + LambertW(1))^2 * LambertW(1)^n). - Vaclav Kotesovec, Feb 18 2023

A383436 a(1) = 1; a(n) = 2 + n * Sum_{k=1..n-1} a(k).

Original entry on oeis.org

1, 4, 17, 90, 562, 4046, 33042, 302098, 3058742, 33986022, 411230866, 5383385882, 75816017838, 1143072268942, 18370804322282, 313528393766946, 5663106612415462, 107932149554271158, 2164639221616216002, 45571352034025600042, 1004848312350264480926, 23159361103691809941342

Offset: 1

Seiichi Manyama, Apr 27 2025

nonn

Cf. A007808, A074143, A082425, A082427, A082428, A082430, A383437.

Cf. A001339.

my(N=30, x='x+O('x^N)); Vec(serlaplace(-2-x/2+(-9*x/2+2*exp(x))/(1-x)^2))

E.g.f.: -2 - x/2 + (-9*x/2 + 2*exp(x))/(1-x)^2.
a(n) = -9*n/2 * n! + 2 * Sum_{k=0..n} (k+1)! * binomial(n,k) for n > 1.
a(n) = (n^2 * a(n-1) - 2)/(n-1) for n > 2.
a(n) = (n+2) * a(n-1) - (n-1) * a(n-2) for n > 3.

A383437 a(1) = 1; a(n) = 5 + n * Sum_{k=1..n-1} a(k).

Original entry on oeis.org

1, 7, 29, 153, 955, 6875, 56145, 513325, 5197415, 57749055, 698763565, 9147450305, 128826591795, 1942308614755, 31215674165705, 532747505761365, 9622751822814655, 183398328858349895, 3678155373214684005, 77434849962414400105, 1707438441671237522315

Offset: 1

Seiichi Manyama, Apr 27 2025

nonn

Cf. A007808, A074143, A082425, A082427, A082428, A082430, A383436.

Cf. A001339.

my(N=30, x='x+O('x^N)); Vec(serlaplace(-5-2*x+(-12*x+5*exp(x))/(1-x)^2))

E.g.f.: -5 - 2*x + (-12*x + 5*exp(x))/(1-x)^2.
a(n) = -12 * n * n! + 5 * Sum_{k=0..n} (k+1)! * binomial(n,k) for n > 1.
a(n) = (n^2 * a(n-1) - 5)/(n-1) for n > 2.
a(n) = (n+2) * a(n-1) - (n-1) * a(n-2) for n > 3.

A144495 Row 2 of array in A144502.

Original entry on oeis.org

2, 7, 30, 155, 946, 6687, 53822, 486355, 4877250, 53759351, 646098622, 8409146187, 117836551730, 1768850337295, 28318532194206, 481652022466307, 8673291031865602, 164849403644999655, 3297954931572397790, 69274457019123638011, 1524368720086682440242

Offset: 0

David Applegate and N. J. A. Sloane, Dec 13 2008

nonn

Robert Israel, Table of n, a(n) for n = 0..447

Cf. A000522, A001339, A038720, A144496, A144497, A144498.

Cf. A144499, A144500, A144501, A144502, A144503.

A144495:= func< n | (&+[Binomial(n,k)*(k+4)*Factorial(k+1) : k in [0..n]])/2 >;
[A144495(n): n in [0..40]]; // G. C. Greubel, Oct 07 2023

f:= rectoproc({a(n)=((4+3*n)*a(n-1)-(n+3)*(n-1)*a(n-2)+(n-1)*(n-2)*a(n-3))/2,a(0)=2,a(1)=7,a(2)=30},a(n),remember):
map(f, [$0..40]); # Robert Israel, Oct 02 2016

(* First program *)
t[0, ] = 1; t[n, 0] := t[n, 0] = t[n-1, 1];
t[n_, k_] := t[n, k] = t[n-1, k+1] + k*t[n, k-1];
a[n_] := t[2, n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Aug 19 2022 *)
(* Second program *)
a[n_]:= a[n]= If[n==0, 2, (n*(n^2+3*n+1)*a[n-1] -(n+2))/(n^2+n-1)];
Table[a[n], {n,0,40}] (* G. C. Greubel, Oct 07 2023 *)

def A144495(n): return sum(binomial(n,j)*factorial(j+1)*(j+4) for j in range(n+1))/2
[A144495(n) for n in range(41)] # G. C. Greubel, Oct 07 2023

E.g.f.: exp(x)*(2-x)/(1-x)^3.
a(n) = (1/2) * (floor((n+1)*(n+1)!*e) + floor(n*n!*e)). [Gary Detlefs, Jun 06 2010]
a(n) = (1/2) * ( A001339(n) + A001339(n+1) ). [Gary Detlefs, Jun 06 2010]
a(n) = (1/2) * (3 + n + (1 + 3*n + n^2) * A000522(n)). - Gerry Martens, Oct 02 2016
a(n) = ((4+3*n)*a(n-1) - (n+3)*(n-1)*a(n-2) + (n-1)*(n-2)*a(n-3))/2. - Robert Israel, Oct 02 2016
From Peter Bala, May 27 2022: (Start)
a(n) = (1/2)*(A000522(n+2) - A000522(n)).
a(n) = (1/2)*Sum_{k = 0..n} binomial(n,k)*(k+4)*(k+1)!; binomial transform of A038720(n+1).
a(n) = (1/2)*e*Integral_{x >= 1} x^n*(x^2 - 1)*exp(-x).
a(2*n) is even and a(2*n+1) is odd. More generally, a(n+k) == a(n) (mod k) for all n and k. It follows that for each positive integer k, the sequence obtained by reducing a(n) modulo k is periodic, with the exact period dividing k. Various divisibility properties of the sequence follow from this; for example, a(3*n+2) == 0 (mod 3), a(5*n+2) == a(5*n+3) (mod 5), a(7*n+1) == 0 (mod 7) and a(11*n+4) == 0 (mod 11). (End)
a(n) = (n*(n^2 + 3*n + 1)*a(n-1) - (n + 2))/(n^2 + n - 1), with a(0) = 2. - G. C. Greubel, Oct 07 2023

A276080 a(n) = A276075(A206296(n)).

Original entry on oeis.org

0, 1, 2, 7, 28, 139, 822, 5677, 44888, 400021, 3966970, 43328131, 516782292, 6682867087, 93130824878, 1391321096089, 22181459914672, 375880800693097, 6746469047955378, 127851581333528191, 2551039715319388940, 53457519928692619411, 1173770856436282074982, 26948387795024752862917, 645694707721735535710728, 16117771962578155161812989

Offset: 0

Antti Karttunen, Aug 18 2016

nonn

Cf. A000045, A000142, A206296, A276075.

Cf. also A276081, A001338, A007188.

A276080 := proc (n) add((n-2*k)*factorial(n-k-1)/factorial(k), k = 0..floor((1/2)*n-1/2)) end proc:
seq(A276080(n), n = 0..25); # Peter Bala, Dec 24 2017

Map[If[# == 1, 0, Total[FactorInteger[#] /. {p_, e_} /; p > 1 :> e PrimePi[p]!]] &, Nest[Append[#, (Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] &@ #[[-1]]) #[[-2]]] &, {1, 2}, 24]] (* Michael De Vlieger, Dec 24 2017 *)

from sympy import factorint, factorial as f, prime, primepi
from operator import mul
from functools import reduce
def a003961(n):
    F=factorint(n)
    return 1 if n==1 else reduce(mul, [prime(primepi(i) + 1)**F[i] for i in F])
def a276075(n):
    F=factorint(n)
    return 0 if n==1 else sum([F[i]*f(primepi(i)) for i in F])
l=[1, 2]
L=[0, 1]
for n in range(2, 11):
    l.append(a003961(l[n - 1])*l[n - 2])
    L.append(a276075(l[n]))
print(L) # Indranil Ghosh, Jun 21 2017

(define (A276080 n) (A276075 (A206296 n)))
;; A more practical standalone program, that uses memoization-macro definec:
(define (A276080 n) (sum_factorials_times_elements_in (A206296as_index_lists n)))
(definec (A206296as_index_lists n) (cond ((zero? n) (list)) ((= 1 n) (list 1)) (else (map + (cons 0 (A206296as_index_lists (- n 1))) (append (A206296as_index_lists (- n 2)) (list 0 0))))))
(define (sum_factorials_times_elements_in nums) (let loop ((s 0) (nums nums) (i 2) (f 1)) (cond ((null? nums) s) (else (loop (+ s (* (car nums) f)) (cdr nums) (+ 1 i) (* i f))))))

a(n) = A276075(A206296(n)).
From Peter Bala, Dec 24 2017: (Start)
a(n) = Sum_{k = 0..floor((n-1)/2)} (n-2*k)!*binomial(n-k-1,k).
O.g.f.: Sum_{n >= 1} n!*x^n/(1 - x^2)^n = x + 2*x^2 + 7*x^3 + 28*x^4 + ....
Cf. A001339(n) = A276075(A007188(n)) for n >= 1, with o.g.f. Sum_{n >= 0} n!*x^n/(1 - x)^n. (End)

cp's OEIS Frontend

A076571 Binomial triangle based on factorials.

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A081923 Expansion of e.g.f.: exp(2x)/(1-x)^2.

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A082427 a(1)=1, a(n) = n * (Sum_{k=1..n-1} a(k)) - 2.

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A082428 a(1) = 1; a(n) = 3 + n * Sum_{k=1..n-1} a(k).

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A269951 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S1(j,k), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.

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Maple

Mathematica

A358738 Expansion of Sum_{k>=0} k! * ( x/(1 - k*x) )^k.

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Mathematica

PARI

PARI

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A383436 a(1) = 1; a(n) = 2 + n * Sum_{k=1..n-1} a(k).

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PARI

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A383437 a(1) = 1; a(n) = 5 + n * Sum_{k=1..n-1} a(k).

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PARI

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A144495 Row 2 of array in A144502.

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A269951 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)C(-j,-n)S1(j,k), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.