A003149 -id:A003149 - cp's OEIS frontend

A309618 a(n) = Sum_{k=0..floor(n/2)} k! * 2^k * (n - 2*k)!.

1, 1, 4, 8, 36, 140, 832, 5376, 42432, 374592, 3720960, 40694784, 486679296, 6310114560, 88168366080, 1320468480000, 21101183631360, 358354687426560, 6444941507297280, 122367252835860480, 2445878526994022400, 51337143210820239360, 1128918790687649955840

Offset: 0

Vaclav Kotesovec, Aug 10 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..448

Cf. A000142, A003149, A107713, A110143, A309619.

Table[Sum[k!*2^k*(n-2*k)!, {k, 0, Floor[n/2]}], {n, 0, 25}]
nmax = 25; CoefficientList[Series[Sum[k!*x^k, {k, 0, nmax}] * Sum[k!*2^k*x^(2 k), {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: B(x)*B(2*x^2), where B(x) is g.f. of A000142.

a(n) ~ n! * (1 + 2/n^2 + 2/n^3 + 10/n^4 + 50/n^5 + 250/n^6 + 1442/n^7 + 9514/n^8 + 68882/n^9 + 539098/n^10 + ...), for coefficients see A326983.

A358446 a(n) = n! * Sum_{k=0..floor(n/2)} 1/binomial(n-k, k).

Original entry on oeis.org

1, 1, 4, 9, 56, 190, 1704, 7644, 93120, 516240, 8136000, 53523360, 1047548160, 7961241600, 187132377600, 1611967392000, 44311886438400, 426483893606400, 13428757601280000, 142790947407360000, 5066854992138240000, 58981696577556480000, 2328441680297779200000

Offset: 0

Vladimir Kruchinin, Nov 16 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A003149, A143216, A344391.

egf := (2*x+1)/((x-1)*(x+1)*(x^2-x-1))-(x*log((1-x)^2*(x+1)))/(-x^2+x+1)^2:
ser := series(egf, x, 22): seq(n!*coeff(ser, x, n), n = 0..20); # Peter Luschny, Nov 17 2022

a(n):=factorial(n)*sum(1/binomial(n-k,k),k,0,floor(n/2));

def A358446(n):
    return sum(A143216(n, k) // A344391(n, k) for k in range((n+2)//2))
print([A358446(n) for n in range(23)]) # Peter Luschny, Nov 17 2022

E.g.f.: (2*x+1)/((x-1)*(x+1)*(x^2-x-1))-(x*log((1-x)^2*(x+1)))/(-x^2+x+1)^2.
a(n) ~ n! * (3 + (-1)^n)/2. - Vaclav Kotesovec, Nov 17 2022
a(n) = Sum_{k=0..floor(n/2)} A143216(n, k)/A344391(n, k). - Peter Luschny, Nov 17 2022

A090319 Fifth column (k=4) of triangle A084938.

Original entry on oeis.org

1, 4, 14, 52, 217, 1040, 5768, 36992, 272584, 2285184, 21550656, 226071744, 2611146384, 32911082496, 449243785728, 6598780563456, 103734755882496, 1737181702840320, 30866291090657280, 579859321408266240

Offset: 0

Philippe Deléham, Feb 05 2004

G. C. Greubel, Table of n, a(n) for n = 0..150

Cf. A084938.

Columns, for k = 0, 1, 2, 3 : A000007, A000142, A003149, A090595.

B:=Binomial;; List([0..20], n-> Sum([0..n], k-> Sum([0..k], m-> Sum([0..m], j-> Factorial(n)/(B(n,k)*B(k,m)*B(m,j)) )))); # G. C. Greubel, Dec 29 2019

F:=Factorial; B:=Binomial; [ (&+[(&+[(&+[F(n)/(B(n,k)*B(k,m)*B(m,j)): j in [0..m]]): m in [0..k]]): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Dec 29 2019

seq( (n+3)!*add(add(add( Beta(k+3,n-k+1)*Beta(m+2,k-m+1)*Beta(j+1,m-j+1), j=0..m), m=0..k), k=0..n), n=0..20); # G. C. Greubel, Dec 29 2019

Table[(n+3)!*Sum[Beta[k+3, n-k+1]*Beta[m+2, k-m+1]*Beta[j+1, m-j+1], {k,0,n}, {m,0,k}, {j,0,m}], {n,0,20}] (* G. C. Greubel, Dec 29 2019 *)

vector(21, n, my(b=binomial); sum(k=0,n-1, sum(m=0,k, sum(j=0,m, (n-1)!/(b(n-1,k)*b(k,m)*b(m,j)) )))) \\ G. C. Greubel, Dec 29 2019

b=binomial; [sum(sum(sum(factorial(n)/(b(n,k)*b(k,m)*b(m,j)) for j in (0..m)) for m in (0..k)) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Dec 29 2019

a(n) = Sum_{k=0..n} A090595(k)*(n-k)!.
a(n) = Sum_{a+b+c+d = n} a!*b!*c!*d!.
a(n) = Sum_{k=0..n} A003149(k)*A003149(n-k).
G.f.: (Sum_{k>=0} k!*x^k)^4.
From G. C. Greubel, Dec 29 2019: (Start)
a(n) = (n+3)!*Sum_{k=0..n} Sum_{m=0..k} Sum_{j=0..m} Beta(k+3, n-k+1)*Beta(m+2, k-m+1)*Beta(j+1, m-j+1), where Beta(x,y) is the Beta function.
a(n) = Sum_{k=0..n} Sum_{m=0..k} Sum_{j=0..m} n!/(binomial(n,k) * binomial(k,m) * binomial(m,j)). (End)

A186374 Number of strong fixed blocks in all the permutations of [n] (see first comment for definition).

Original entry on oeis.org

0, 1, 1, 3, 11, 48, 248, 1500, 10476, 83328, 745344, 7413120, 81187200, 970928640, 12589240320, 175900757760, 2634526944000, 42103369728000, 715107004416000, 12862666543104000, 244249409359872000, 4882687056543744000, 102496533840691200000

Offset: 0

Emeric Deutsch, Apr 18 2011

nonn

A fixed block of a permutation p is a maximal sequence of consecutive fixed points of p. For example, the permutation 213486759 has 3 fixed blocks: 34, 67, and 9. A fixed block f of a permutation p is said to be strong if all the entries to the left (right) of f are smaller (larger) than all the entries of f. In the above example, only 34 and 9 are strong fixed blocks.

			a(3) = 3 because in [123], [1]32, 21[3], 231, 312, 321 we have 1 + 1 + 1 + 0 + 0 + 0 strong fixed blocks (shown between square brackets).

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A186373, A003149.

a:= proc(n) option remember; `if`(n<5, [0, 1, 1, 3, 11][n+1],
      ((3*n^2-12*n+2)*a(n-1) -(n^3-3*n^2-8*n+23)*a(n-2)
       +(n-3)^3*a(n-3)) / (2*n-8))
    end:
seq(a(n), n=0..24);  # Alois P. Heinz, May 22 2013

Flatten[{0, 1, Table[(n-1)! + Sum[k!*(n-2-k)!*(n-2-k), {k,0,n-2}], {n,2,20}]}] (* Vaclav Kotesovec, Aug 04 2015 *)
Flatten[{0, Simplify[Table[Gamma[n] * (1 - (n-2)*(I*Pi/2^n + LerchPhi[2, 1, n])), {n, 1, 20}]]}] (* Vaclav Kotesovec, Aug 04 2015 *)

a(n) = Sum(k*A186373(n,k), k>=0).
Apparently, a(n) = A003149(n-1)-A003149(n-2) or, equivalently, a(n)=(n-1)! + Sum(k!*(n-2-k)!*(n-2-k), k=0..n-2).
a(n) ~ 2 * (n-1)! * ((1 + 1/n^2 + 7/n^3 + 49/n^4 + 391/n^5 + 3601/n^6 + 37927/n^7 + 451249/n^8 + 5995591/n^9 + 88073041/n^10)). - Vaclav Kotesovec, Mar 17 2015
Recurrence (for n>=3): 2*(n^2 - 7*n + 11)*a(n) = (n-2)*(3*n^2 - 17*n + 17)*a(n-1) - (n-2)^2*(n^2 - 5*n + 5)*a(n-2). - Vaclav Kotesovec, Aug 04 2015

a(11)-a(22) from Alois P. Heinz, May 22 2013

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.

A333370 Convolution of primorial numbers (A002110) with themselves.

Original entry on oeis.org

1, 4, 16, 84, 576, 5820, 72720, 1181460, 21984480, 493882620, 13996733520, 430612001820, 15742074348000, 641147559872820, 27488197348531920, 1286344285877911260, 67817877972050366160, 3984226025421591129180, 242703493548359285922480, 16211176424801583698573100

Offset: 0

Ilya Gutkovskiy, Mar 17 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..350
Eric Weisstein's World of Mathematics, Primorial
Index entries for sequences related to primorial numbers

Cf. A002110, A003149, A014342, A333371.

p:= proc(n) option remember; `if`(n<1, 1, ithprime(n)*p(n-1)) end:
a:= n-> add(p(i)*p(n-i), i=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 17 2020

primorial[n_] := Product[Prime[k], {k, 1, n}]; a[n_] := Sum[primorial[k] primorial[n - k], {k, 0, n}]; Table[a[n], {n, 0, 19}]

G.f.: (Sum_{k>=0} prime(k)# * x^k)^2, where prime()# = A002110.

a(n) = Sum_{k=0..n} prime(k)# * prime(n-k)#.

A357240 Expansion of e.g.f. 2 * (exp(x) - 1) / (exp(exp(x) - 1) + 1).

Original entry on oeis.org

0, 1, 0, -2, -5, -4, 32, 225, 794, 190, -22291, -200298, -920244, 924223, 65848880, 716920754, 3831260555, -13147083976, -575844827780, -7162425813919, -40755845041730, 320194436283162, 11810647258173653, 161108090793013130, 896865861205240824, -14305712791762925929, -487306962045115504436

Offset: 0

Ilya Gutkovskiy, Sep 19 2022

sign

Stirling transform of the Genocchi numbers (of first kind, A036968).

Alois P. Heinz, Table of n, a(n) for n = 0..494

Cf. A001469, A003149, A036968, A059371.

b:= proc(n, m) option remember; `if`(n=0, `if`(m=0, 0,
      m*euler(m-1, 0)), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..26);  # Alois P. Heinz, Jun 23 2023

nmax = 26; CoefficientList[Series[2 (Exp[x] - 1)/(Exp[Exp[x] - 1] + 1), {x, 0, nmax}], x] Range[0, nmax]!
Table[2 Sum[StirlingS2[n, k] (1 - 2^k) BernoulliB[k], {k, 0, n}], {n, 0, 26}]

a(n) = 2*sum(k=0, n, stirling(n, k, 2)*(1-2^k)*bernfrac(k)); \\ Michel Marcus, Sep 20 2022

a(n) = 2 * Sum_{k=0..n} Stirling2(n,k) * (1 - 2^k) * Bernoulli(k).

a(n) ~ Pi^(3/2) * 2^(n + 7/2) * n^(n + 1/2) * (cos(n*arctan(2*arctan(Pi)/log(1 + Pi^2))) * (Pi*log(1 + Pi^2) + 2*arctan(Pi)) + (log(1 + Pi^2) - 2*Pi*arctan(Pi)) * sin(n*arctan(2*arctan(Pi)/log(1 + Pi^2)))) / ((1 + Pi^2) * exp(n) * (4*arctan(Pi)^2 + log(1 + Pi^2)^2)^(n/2 + 1)). - Vaclav Kotesovec, Oct 04 2022

A173476 Triangle T(n, k) = 1 + (k!)^2 - 2k!(n-k)! + ((n-k)!)^2, read by rows.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 26, 2, 2, 26, 530, 26, 1, 26, 530, 14162, 530, 17, 17, 530, 14162, 516962, 14162, 485, 1, 485, 14162, 516962, 25391522, 516962, 13925, 325, 325, 13925, 516962, 25391522, 1625621762, 25391522, 515525, 12997, 1, 12997, 515525, 25391522, 1625621762

Offset: 0

Roger L. Bagula, Feb 19 2010

			Triangle begins as:
           1;
           1,        1;
           2,        1,      2;
          26,        2,      2,    26;
         530,       26,      1,    26, 530;
       14162,      530,     17,    17, 530, 14162;
      516962,    14162,    485,     1, 485, 14162, 516962;
    25391522,   516962,  13925,   325, 325, 13925, 516962, 25391522;
  1625621762, 25391522, 515525, 12997,   1, 12997, 515525, 25391522, 1625621762;

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A003149, A061062.

[(Factorial(n-k) -Factorial(k))^2 +1: k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 19 2021

Table[((n-k)! -k!)^2 +1, {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Feb 19 2021 *)

flatten([[(factorial(n-k) -factorial(k))^2 +1 for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 19 2021

T(n, k) = 1 + ( (n-k)! - k! )^2.

Sum_{k=0..n} T(n, k) = 1 + n + 2*A061062(n) - 2*A003149(n). - G. C. Greubel, Feb 19 2021

Edited by G. C. Greubel, Feb 19 2021

A235802 Expansion of e.g.f.: 1/(1 - x)^(2/(2-x)).

Original entry on oeis.org

1, 1, 3, 12, 61, 375, 2697, 22176, 204977, 2102445, 23685615, 290642220, 3857751573, 55063797243, 840956549517, 13682498891040, 236257301424225, 4314883836968505, 83102361300891963, 1683252077760375660, 35770269996769203405, 795749735451309432255

Offset: 0

Paul D. Hanna, Jan 15 2014

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)) = x*(1) + x^2/2*(1 + 1) + x^3/3*(1 + 1/2 + 1) + x^4/4*(1 + 1/3 + 1/3 + 1) + x^5/5*(1 + 1/4 + 1/6 + 1/4 + 1) + x^6/6*(1 + 1/5 + 1/10 + 1/10 + 1/5 + 1) + ...
Explicitly, the logarithm begins:
log(A(x)) = x + 2*x^2/2! + 5*x^3/3! + 16*x^4/4! + 64*x^5/5! + 312*x^6/6! + 1812*x^7/7! + 12288*x^8/8! + ... + A003149(n-1)*x^n/n! + ...

G. C. Greubel, Table of n, a(n) for n = 0..430

Cf. A003149, A193425.

R:=PowerSeriesRing(Rationals(), 50); Coefficients(R!(Laplace( 1/(1-x)^(2/(2-x)) ))); // G. C. Greubel, Jul 12 2023

CoefficientList[Series[1/(1-x)^(2/(2-x)), {x,0,20}], x]*Range[0,20]! (* Vaclav Kotesovec, Jul 13 2014 *)

{a(n)=n!*polcoeff(exp(sum(m=1, n, x^m/m*sum(k=0, m-1, 1/binomial(m-1, k))) +x*O(x^n)), n)}
for(n=0,25,print1(a(n),", "))

{a(n)=n!*polcoeff(1/(1-x+x*O(x^n))^(2/(2-x)), n)}
for(n=0,25,print1(a(n),", "))

m=50
def f(x): return exp(sum(sum( 1/binomial(n-1,k) for k in range(n))*x^n/n for n in range(1,m+2)))
def A235802_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).egf_to_ogf().list()
A235802_list(m) # G. C. Greubel, Jul 12 2023

E.g.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n-1} 1/C(n-1,k) ).
E.g.f.: exp( Sum_{n>=1} A003149(n-1)*x^n/n! ), where A003149(n) = Sum_{k=0..n} k!*(n-k)!.
a(n) ~ n! * (n-2*log(n)). - Vaclav Kotesovec, Jul 13 2014

A279055 Self-convolution of squares of factorial numbers (A001044).

Original entry on oeis.org

1, 2, 9, 80, 1240, 30240, 1071504, 51996672, 3307723776, 266872320000, 26615381760000, 3214252921651200, 462189467175321600, 78024380924038348800, 15279632043682406400000, 3435553774431004262400000, 879010223384483132866560000, 253916900613208108255150080000

Offset: 0

Arman Maesumi, Dec 04 2016

nonn

a(n) = (n!)^2 * Sum_{i=0..n} (binomial(n,i)^(-2)).
Consider a triangle ABC with area p. Let points X, Y, Z be randomly and uniformly chosen on sides BC, CA, BA. Let r = area of XYZ. Then the average or expected value of (r/p)^n = a(n)/(n!^2 * (n+1)^3).
a(n) = (3*(n+1)^4 *(n!)^4 /(2n+3)!) * Sum_{i=1..n+1} ((1/i)* binomial(2i, i)), see Sprugnoli Formula 5.2 as noted by Markus Scheuer.

Seiichi Manyama, Table of n, a(n) for n = 0..253
Arman Maesumi, Triangle Inscribed-Triangle Picking, arXiv:1804.11007 [math.GM], 2018.
R. Sprugnoli, Riordan Array Proofs of Identities in Gould's Book.

Cf. A000142, A001044, A003149, A100516, A100517.

Table[Sum[(k!*(n-k)!)^2, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 05 2016 *)

a(n) = Sum_{i=0..n} (i! * (n-i)!)^2.
a(n) ~ 2*(n!)^2. - Vaclav Kotesovec, Dec 05 2016
a(n) = A001044(n)*A100516(n)/A100517(n). - Alois P. Heinz, Feb 21 2023

Definition clarified by Georg Fischer, Feb 21 2023

cp's OEIS Frontend

A309618 a(n) = Sum_{k=0..floor(n/2)} k! * 2^k * (n - 2*k)!.

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A358446 a(n) = n! * Sum_{k=0..floor(n/2)} 1/binomial(n-k, k).

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A090319 Fifth column (k=4) of triangle A084938.

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A186374 Number of strong fixed blocks in all the permutations of [n] (see first comment for definition).

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

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A333370 Convolution of primorial numbers (A002110) with themselves.

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A357240 Expansion of e.g.f. 2 * (exp(x) - 1) / (exp(exp(x) - 1) + 1).

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A173476 Triangle T(n, k) = 1 + (k!)^2 - 2k!(n-k)! + ((n-k)!)^2, read by rows.