A126674 -id:A126674 - cp's OEIS frontend

A126671 Triangle read by rows: row n (n>=0) has g.f. Sum_{i=1..n} n!x^i(1+x)^(n-i)/(n+1-i).

0, 0, 1, 0, 1, 3, 0, 2, 7, 11, 0, 6, 26, 46, 50, 0, 24, 126, 274, 326, 274, 0, 120, 744, 1956, 2844, 2556, 1764, 0, 720, 5160, 16008, 28092, 30708, 22212, 13068, 0, 5040, 41040, 147120, 304464, 401136, 351504, 212976, 109584, 0, 40320

Offset: 1

N. J. A. Sloane and Carlo Wood (carlo(AT)alinoe.com), Feb 13 2007

The first nonzero column gives the factorial numbers, which are Stirling_1(*,1), the rightmost diagonal gives Stirling_1(*,2), so this triangle may be regarded as interpolating between the first two columns of the Stirling numbers of the first kind.
This is a slice (the right-hand wall) through the infinite square pyramid described in the link. The other three walls give A007318 and A008276 (twice).
The coefficients of the A165674 triangle are generated by the asymptotic expansion of the higher order exponential integral E(x,m=2,n). The a(n) formulas for the coefficients in the right hand columns of this triangle lead to Wiggen's triangle A028421 and their o.g.f.s. lead to the sequence given above. Some right hand columns of the A165674 triangle are A080663, A165676, A165677, A165678 and A165679. - Johannes W. Meijer, Oct 07 2009

			Triangle begins:
0,
0, 1,
0, 1, 3,
0, 2, 7, 11,
0, 6, 26, 46, 50,
0, 24, 126, 274, 326, 274,
0, 120, 744, 1956, 2844, 2556, 1764,
0, 720, 5160, 16008, 28092, 30708, 22212, 13068,
0, 5040, 41040, 147120, 304464, 401136, 351504, 212976, 109584,
0, 40320, 367920, 1498320, 3582000, 5562576, 5868144, 4292496, 2239344, 1026576, ...

N. J. A. Sloane, Notes on Carlo Wood's Polynomials

Columns give A000142, A108217, A126672; diagonals give A000254, A067318, A126673. Row sums give A126674. Alternating row sums give A000142.
See A126682 for the full pyramid of coefficients of the underlying polynomials.
Cf. A165674, A028421, A080663, A165676, A165677, A165678 and A165679. - Johannes W. Meijer, Oct 07 2009

for n from 1 to 15 do t1:=add( n!*x^i*(1+x)^(n-i)/(n+1-i), i=1..n); series(t1,x,100); lprint(seriestolist(%)); od:

Join[{{0}}, Reap[For[n = 1, n <= 15, n++, t1 = Sum[n!*x^i*(1+x)^(n-i)/(n+1-i), {i, 1, n}]; se = Series[t1, {x, 0, 100}]; Sow[CoefficientList[se, x]]]][[2, 1]]] // Flatten (* Jean-François Alcover, Jan 07 2014, after Maple *)

Recurrence: T(n,0) = 0; for n>=0, i>=1, T(n+1,i) = (n+1)*T(n,i) + n!*binomial(n,i).

E.g.f.: x*log(1-(1+x)*y)/(x*y-1)/(1+x). - Vladeta Jovovic, Feb 13 2007

A193425 Expansion of e.g.f.: (1 - 2*x)^(-1/(1-x)).

Original entry on oeis.org

1, 2, 12, 96, 976, 12000, 172608, 2838528, 52474112, 1076451840, 24254069760, 595235266560, 15801350443008, 451082627014656, 13778232107286528, 448348123661598720, 15483358506138009600, 565560454279135887360

Offset: 0

Paul D. Hanna, Jul 27 2011

nonn

			E.g.f.: A(x) = 1 + 2*x + 12*x^2/2! + 96*x^3/3! + 976*x^4/4! + 12000*x^5/5! +...
where the logarithm involves sums of reciprocal binomial coefficients:
log(A(x)) = 2*x*(1) + (2*x)^2/2*(1 + 1) + (2*x)^3/3*(1 + 1/2 + 1) + (2*x)^4/4*(1 + 1/3 + 1/3 + 1) + (2*x)^5/5*(1 + 1/4 + 1/6 + 1/4 + 1) + (2*x)^6/6*(1 + 1/5 + 1/10 + 1/10 + 1/5 + 1) +...
Explicitly, the logarithm begins:
log(A(x)) = 2*x + 8*x^2/2! + 40*x^3/3! + 256*x^4/4! + 2048*x^5/5! + 19968*x^6/6! +...
in which the coefficients equal 2*A126674(n).

G. C. Greubel, Table of n, a(n) for n = 0..400

Cf. A126674.

m:=50;
f:= func< x | Exp((&+[(&+[ 1/Binomial(n-1,k): k in [0..n-1]])*(2*x)^n/n: n in [1..m+2]])) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!(Laplace( f(x) ))); // G. C. Greubel, Feb 02 2023

CoefficientList[Series[(1-2*x)^(-1/(1-x)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 27 2013 *)

{a(n)=n!*polcoeff(exp(sum(m=1,n,2^m*x^m/m*sum(k=0,m-1,1/binomial(m-1,k)))+x*O(x^n)),n)}

{a(n)=n!*polcoeff((1-2*x+x*O(x^n))^(-1/(1-x)),n)}

m=50
def f(x): return exp(sum(sum( 1/binomial(n-1,k) for k in range(n))*(2*x)^n/n for n in range(1,m+2)))
def A193425_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).egf_to_ogf().list()
A193425_list(m) # G. C. Greubel, Feb 02 2023

E.g.f.: exp( Sum_{n>=1} (2*x)^n/n * Sum_{k=0..n-1} 1/C(n-1,k) ).
E.g.f.: exp( Sum_{n>=1} 2*A126674(n)*x^n/n ), where A126674(n) = n!*Sum_{j=0..n-1} 2^j/(j+1).
a(n) ~ n!*n*2^n * (1 - 2*log(n)/n). - Vaclav Kotesovec, Jun 27 2013

A293471 a(n) = [x^n] (1/(1 - 2x/(1 - 2x/(1 - 4x/(1 - 4x/(1 - 6x/(1 - 6x/(1 - ...))))))))^n, a continued fraction.

Original entry on oeis.org

1, 2, 20, 248, 3472, 53152, 878144, 15577984, 296411392, 6054973952, 132994708480, 3144712222720, 80063883022336, 2192452931723264, 64427309553434624, 2025284853319303168, 67859418068644069376, 2414526405567056052224, 90909088845844899430400, 3610058425696043667030016

Offset: 0

Ilya Gutkovskiy, Oct 09 2017

nonn

Index entries for sequences related to factorial numbers

Cf. A000165, A052712, A126674, A287899, A293470.

Table[SeriesCoefficient[1/(1 + ContinuedFractionK[-2 Floor[(k + 1)/2] x, 1, {k, 1, n}])^n, {x, 0, n}], {n, 0, 19}]
Table[SeriesCoefficient[Sum[(2 k)!! x^k, {k, 0, n}]^n, {x, 0, n}], {n, 0, 19}]

a(n) ~ sqrt(Pi) * 2^(n + 1/2) * n^(n + 3/2) / exp(n-1). - Vaclav Kotesovec, Sep 16 2021

A305577 a(n) = Sum_{k=0..n} k!!*(n - k)!!.

Original entry on oeis.org

1, 2, 5, 10, 26, 58, 167, 414, 1324, 3606, 12729, 37674, 145578, 463770, 1944879, 6614190, 29852856, 107616150, 518782545, 1970493210, 10077228270, 40125873690, 216425656215, 899557170750, 5091758227620, 22011865939350, 130202223160905, 583641857191050, 3594820517111250

Offset: 0

Ilya Gutkovskiy, Jun 05 2018

nonn

Convolution of A006882 with itself.

Alois P. Heinz, Table of n, a(n) for n = 0..807
Poloni, Federico; Del Corso, Gianna M. Counting Fiedler pencils with repetitions. Linear Algebra Appl. 532, 463-499 (2017), corollary 24.
Eric Weisstein's World of Mathematics, Double Factorial
Index entries for sequences related to factorial numbers

Cf. A003149, A006882, A034430, A059371, A111308, A126674, A305578.

a:= proc(n) option remember; `if`(n<4, n^2+1,
      ((3*n^2-4*n-2)*a(n-2) +(n+1)*a(n-3)
       -2*a(n-1) -(n-1)^2*n*a(n-4))/(2*n-4))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Jun 14 2018

Table[Sum[k!! (n - k)!!, {k, 0, n}], {n, 0, 28}]
nmax = 28; CoefficientList[Series[Sum[k!! x^k, {k, 0, nmax}]^2, {x, 0, nmax}], x]

G.f.: (Sum_{k>=0} k!!*x^k)^2.

A384200 Expansion of e.g.f. -log(1 - 3x)/(3 (1 - x)).

Original entry on oeis.org

0, 1, 5, 33, 294, 3414, 49644, 872388, 18001584, 426553776, 11408104800, 339766164000, 11148335337600, 399489448694400, 15520734764640000, 649782085752172800, 29160211264750540800, 1396381090351116441600, 71068392067688315596800, 3830710201119961857331200

Offset: 0

Seiichi Manyama, May 22 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A126674.

[0] cat [n le 1 select 1  else  n * Self(n-1) + 3^(n-1) * Factorial(n-1): n in [1..20]]; // Vincenzo Librandi, May 22 2025

a[n_]:= n! * Sum[(3)^(k-1)/k,{k,1,n}];Table[a[n],{n,0,19}] (* Vincenzo Librandi, May 22 2025 *)

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

a(n) = n! * Sum_{k=1..n} 3^(k-1)/k.
a(n) = n * a(n-1) + 3^(n-1) * (n-1)!.
a(n) = (4*n-3) * a(n-1) - 3 * (n-1)^2 * a(n-2).

cp's OEIS Frontend

A126671 Triangle read by rows: row n (n>=0) has g.f. Sum_{i=1..n} n!*x^i*(1+x)^(n-i)/(n+1-i).

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A193425 Expansion of e.g.f.: (1 - 2*x)^(-1/(1-x)).

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A293471 a(n) = [x^n] (1/(1 - 2*x/(1 - 2*x/(1 - 4*x/(1 - 4*x/(1 - 6*x/(1 - 6*x/(1 - ...))))))))^n, a continued fraction.

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A305577 a(n) = Sum_{k=0..n} k!!*(n - k)!!.

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Maple

Mathematica

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A384200 Expansion of e.g.f. -log(1 - 3*x)/(3 * (1 - x)).

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Magma

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Formula

A126671 Triangle read by rows: row n (n>=0) has g.f. Sum_{i=1..n} n!x^i(1+x)^(n-i)/(n+1-i).

A293471 a(n) = [x^n] (1/(1 - 2x/(1 - 2x/(1 - 4x/(1 - 4x/(1 - 6x/(1 - 6x/(1 - ...))))))))^n, a continued fraction.

A384200 Expansion of e.g.f. -log(1 - 3x)/(3 (1 - x)).