A052517 -id:A052517 - cp's OEIS frontend

A101612 n! * Sum[k=floor(n/2)..n, 1/k].

3, 11, 26, 154, 684, 5508, 35664, 361296, 3068640, 37383840, 392722560, 5584394880, 69878833920, 1135360800000, 16484477184000, 301158902016000, 4976250951168000, 100951141777920000, 1870345490614272000

Offset: 2

Ralf Stephan, Dec 10 2004

Cf. A101609, A101610, A101611, A101613, A000254, A046673, A052517.

A052767 Expansion of e.g.f.: -(log(1-x))^5.

Original entry on oeis.org

0, 0, 0, 0, 0, 120, 1800, 21000, 235200, 2693880, 32319000, 410031600, 5519487600, 78864820320, 1194924450720, 19166592681600, 324817601472000, 5803921108010880, 109115988701293440, 2154085473710580480, 44566174481427360000, 964537418717406213120, 21799797542483649131520

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Andrew Howroyd, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 724

Column k=5 of A225479.

Cf. A000482, A052517.

spec := [S,{B=Cycle(Z),S=Prod(B,B,B,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=20},CoefficientList[Series[-(Log[1-x])^5,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 14 2019 *)

a(n) = {5!*stirling(n,5,1)*(-1)^(n+1)} \\ Andrew Howroyd, Jul 27 2020

E.g.f.: log(-1/(-1+x))^5.
Recurrence: a(1)=0, a(0)=0, a(2)=0, a(4)=0, a(3)=0, (-1-5*n-10*n^2-10*n^3-5*n^4-n^5)*a(n+1) + (31+5*n^4+70*n^2+30*n^3+75*n)*a(n+2) + (-125*n-90-60*n^2-10*n^3)*a(n+3) + (10*n^2+65+50*n)*a(n+4) + (-15-5*n)*a(n+5) + a(n+6)=0, a(5)=120.
a(n) = 120*A000482(n) = 5!*Stirling1(n,5)*(-1)^(n+1). - Andrew Howroyd, Jul 27 2020

Definition clarified by Harvey P. Dale, Oct 14 2019

Terms a(20) and beyond from Andrew Howroyd, Jul 27 2020

A052779 Expansion of e.g.f.: (log(1-x))^6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 720, 15120, 231840, 3265920, 45556560, 649479600, 9604465200, 148370508000, 2402005525920, 40797624067200, 726963917097600, 13580328282393600, 265689107448756480, 5437099866285377280, 116229410301685651200, 2591985252922277184000, 60218914823672258142720

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Original name: a simple grammar.

Andrew Howroyd, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 736

Column k=6 of A225479.

Cf. A001233, A052517.

spec := [S,{B=Cycle(Z),S=Prod(B,B,B,B,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

a(n) = {6!*stirling(n,6,1)*(-1)^n} \\ Andrew Howroyd, Jul 27 2020

E.g.f.: log(-1/(-1+x))^6.
Recurrence: {a(1)=0, a(0)=0, a(2)=0, a(4)=0, a(3)=0, a(5)=0, a(6)=720, (1+15*n^2+6*n+6*n^5+15*n^4+20*n^3+n^6)*a(n+1) + (-63-186*n-225*n^2-6*n^5-45*n^4-140*n^3)*a(n+2) + (540*n+120*n^3+375*n^2+15*n^4+301)*a(n+3) + (-390*n-20*n^3-350-150*n^2)*a(n+4) + (140+15*n^2+90*n)*a(n+5) + (-21-6*n)*a(n+6) + a(n+7)}.
a(n) = 720*A001233(n) = 6!*(-1)^n*Stirling1(n,6). - Andrew Howroyd, Jul 27 2020

Name changed and terms a(20) and beyond from Andrew Howroyd, Jul 27 2020

A129841 Antidiagonal sums of triangle T defined in A048594: T(j,k) = k! * Stirling1(j,k), 1<= k <= j.

Original entry on oeis.org

1, -1, 4, -12, 52, -256, 1502, -10158, 78360, -680280, 6574872, -70075416, 816909816, -10342968456, 141357740736, -2074340369088, 32530886655168, -542971977209760, 9610316495698416, -179788450082431536, 3544714566466060032

Offset: 1

Paul Curtz, May 22 2007

sign

			First seven rows of T are
[    1 ]
[   -1,      2 ]
[    2,     -6,      6 ]
[   -6,     22,    -36,     24 ]
[   24,   -100,    210,   -240,    120 ]
[ -120,    548,  -1350,   2040,  -1800,    720 ]
[  720,  -3528,   9744, -17640,  21000, -15120,   5040 ]

P. Curtz, Integration numerique des systemes differentiels a conditions initiales. Note no. 12 du Centre de Calcul Scientifique de l'Armement, 1969, 135 pages, p. 61. Available from Centre d'Electronique de L'Armement, 35170 Bruz, France, or INRIA, Projets Algorithmes, 78150 Rocquencourt.
P. Curtz, Gazette des Mathematiciens, 1992, no. 52, p. 44.
P. Flajolet, X. Gourdon and B. Salvy, Gazette des Mathematiciens, 1993, no. 55, pp. 67-78.

Cf. A048594 (T read by rows), A075181 (T unsigned with rows read backwards), A006252 (row sums of T), A000142 (main diagonal of T), A001286 (unsigned first subdiagonal of T). Unsigned values of second through sixth column of T are in A052517, A052748, A052753, A052767, A052779 resp.

m:=21; T:=[ [ Factorial(k)*StirlingFirst(j, k): k in [1..j] ]: j in [1..m] ]; [ &+[ T[j-k+1][k]: k in [1..(j+1) div 2] ]: j in [1..m] ]; // Klaus Brockhaus, Jun 03 2007

m = 21; t[j_, k_] := k!*StirlingS1[j, k]; Total /@ Table[ t[j-k+1, k], {j, 1, m}, {k, 1, Quotient[j+1, 2]}] (* Jean-François Alcover, Aug 13 2012, translated from Klaus Brockhaus's Magma program *)

E.g.f. for k-th column (k>=1): log(1+x)^k. For further formulas see the references.

Edited and extended by Klaus Brockhaus, Jun 03 2007

A304654 a(n) = (n!)^2 * Sum_{k=1..n-1} 1/(k^2*(n-k)).

Original entry on oeis.org

0, 0, 4, 27, 328, 6500, 192216, 7952112, 438941952, 31185057024, 2772643115520, 301622403456000, 39413353102848000, 6091955683706880000, 1099401414283210752000, 229088914497045356544000, 54589580461769879715840000, 14750581694440372638842880000

Offset: 0

Vaclav Kotesovec, May 16 2018

nonn

Cf. A052517, A302827, A304589, A304655.

Table[(n!)^2 * Sum[1/(k^2*(n-k)), {k, 1, n-1}], {n, 0, 20}]

Recurrence: (2*n - 3)*a(n) = (6*n^3 - 25*n^2 + 33*n - 12)*a(n-1) - (n-2)^2*(6*n^3 - 29*n^2 + 42*n - 15)*a(n-2) + (n-3)^3*(n-2)^3*(2*n - 1)*a(n-3).

a(n)/(n!)^2 ~ Pi^2/(6*n).

A304655 a(n) = (n!)^3 * Sum_{k=1..n-1} 1/(k^3*(n-k)^2).

Original entry on oeis.org

0, 0, 8, 81, 2480, 175000, 23825904, 5563712448, 2051674085376, 1124193889529856, 873600549068759040, 927968580453961728000, 1307864687259363065856000, 2386263863328126193631232000, 5521179117888960788194394112000, 15917227342113559040727019683840000

Offset: 0

Vaclav Kotesovec, May 16 2018

nonn

Cf. A052517, A302827, A304589, A304654.

Table[(n!)^3 * Sum[1/(k^3*(n-k)^2), {k, 1, n-1}], {n, 0, 20}]

Recurrence: n*(12*n^4 - 108*n^3 + 354*n^2 - 501*n + 260)*a(n) = 2*(n-1)*(24*n^7 - 306*n^6 + 1620*n^5 - 4599*n^4 + 7516*n^3 - 7015*n^2 + 3444*n - 696)*a(n-1) - 6*(n-2)^4*(12*n^7 - 162*n^6 + 906*n^5 - 2700*n^4 + 4583*n^3 - 4378*n^2 + 2163*n - 436)*a(n-2) + 2*(n-3)^4*(n-2)^3*(24*n^7 - 342*n^6 + 2004*n^5 - 6201*n^4 + 10816*n^3 - 10497*n^2 + 5208*n - 1048)*a(n-3) - (n-4)^5*(n-3)^5*(n-2)^3*(12*n^4 - 60*n^3 + 102*n^2 - 69*n + 17)*a(n-4).

a(n)/(n!)^3 ~ Zeta(3)/n^2.

A308346 Expansion of e.g.f. 1/(1 - x)^log(1 - x).

Original entry on oeis.org

1, 0, -2, -6, -10, 20, 352, 2772, 18132, 104400, 469608, 238920, -35811048, -730972944, -11436661728, -164609993520, -2294024595312, -31488879303552, -426338226719904, -5626751283423072, -70000948158061728, -745703905072996800, -4142683990211677440, 110386551348875714880

Offset: 0

Ilya Gutkovskiy, May 21 2019

Robert Israel, Table of n, a(n) for n = 0..453

Cf. A000254, A008405, A009199, A052517, A067994, A086660, A087761.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (1-x)^Log(1/(1-x)) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 21 2019

E:= 1/(1-x)^log(1-x):
S:= series(E,x,31):
seq(coeff(S,x,j)*j!,j=0..30); # Robert Israel, May 22 2019

nmax = 23; CoefficientList[Series[1/(1 - x)^Log[1 - x], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] HermiteH[k, 0], {k, 0, n}], {n, 0, 23}]
a[n_] := a[n] = -2 Sum[(k - 1)! HarmonicNumber[k - 1] Binomial[n - 1, k - 1] a[n - k], {k, 2, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]

a(n) = sum(k=0, n, abs(stirling(n, k, 1))*polhermite(k, 0)); \\ Michel Marcus, May 21 2019

m = 30; T = taylor((1-x)^log(1/(1-x)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 21 2019

a(n) = Sum_{k=0..n} |Stirling1(n,k)|*A067994(k).

A333371 Exponential convolution of primorial numbers (A002110) with themselves.

Original entry on oeis.org

1, 4, 20, 132, 1116, 12420, 171300, 2884980, 56674380, 1289511300, 34769949060, 1063909626780, 37255008811020, 1470406699982220, 63114539746598340, 2936218980067393020, 150241360192861037100, 8497891914008911514100, 514514062115556069627060

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, A014345, A052517, A062119, A136104, A333370.

p:= proc(n) option remember; `if`(n<1, 1, ithprime(n)*p(n-1)) end:
a:= n-> add(p(i)*p(n-i)*binomial(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[Binomial[n, k] primorial[k] primorial[n - k], {k, 0, n}]; Table[a[n], {n, 0, 18}]

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

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

A156047 Triangle read by rows: T(n, k) = (n+1)!*(1/k + 1/(n-k+1)).

Original entry on oeis.org

4, 9, 9, 32, 24, 32, 150, 100, 100, 150, 864, 540, 480, 540, 864, 5880, 3528, 2940, 2940, 3528, 5880, 46080, 26880, 21504, 20160, 21504, 26880, 46080, 408240, 233280, 181440, 163296, 163296, 181440, 233280, 408240, 4032000, 2268000, 1728000, 1512000, 1451520, 1512000, 1728000, 2268000, 4032000

Offset: 1

Roger L. Bagula, Feb 02 2009

Row sums are (n+1)*A052517(n+2) = {4, 18, 88, 500, 3288, 24696, 209088, 1972512, 20531520, ...}.

			Triangle begins as:
      4;
      9,     9;
     32,    24,    32;
    150,   100,   100,   150;
    864,   540,   480,   540,   864;
   5880,  3528,  2940,  2940,  3528,  5880;
  46080, 26880, 21504, 20160, 21504, 26880, 46080;

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A001008, A002805, A052517, A058298.

Flat(List([1..10], n-> List([1..n], k-> (n+1)*Factorial(n+1)/(k*(n-k+1)) ))); # G. C. Greubel, Dec 02 2019

[(n+1)*Factorial(n+1)/(k*(n-k+1)): k in [1..n], n in [1..10]]; // G. C. Greubel, Dec 02 2019

seq(seq( (n+1)*(n+1)!/(k*(n-k+1)), k=1..n), n=1..10); # G. C. Greubel, Dec 02 2019

Table[(n+1)*(n+1)!/(k*(n-k+1)), {n,10}, {k,n}]//Flatten (* modified by G. C. Greubel, Dec 02 2019 *)

T(n,k) = (n+1)*(n+1)!/(k*(n-k+1)); \\ G. C. Greubel, Dec 02 2019

[[(n+1)*factorial(n+1)/(k*(n-k+1)) for k in (1..n)] for n in (1..10)] # G. C. Greubel, Dec 02 2019

T(n, k) = (n+1)*(n+1)!/(k*(n-k+1)).

Sum_{k=1..n} T(n,k) = 2*(n+1)!*H(n), where H(n) is the harmonic number. - G. C. Greubel, Dec 02 2019

Offset changed by G. C. Greubel, Dec 02 2019

A377394 Expansion of e.g.f. (1 - log(1-x))^3.

Original entry on oeis.org

1, 3, 9, 30, 120, 582, 3354, 22488, 172320, 1487208, 14284296, 151179696, 1748521296, 21945019392, 297077918976, 4315269544704, 66952906801920, 1105127533048320, 19337110495511040, 357542547031249920, 6965984564179246080, 142638952766943744000, 3062533108375448064000

Offset: 0

Seiichi Manyama, Oct 27 2024

Seiichi Manyama, Table of n, a(n) for n = 0..450

Cf. A052517, A052748.

a(n) = sum(k=0, 3, k!*binomial(3, k)*abs(stirling(n, k, 1)));

a(n) = Sum_{k=0..3} k! * binomial(3,k) * |Stirling1(n,k)|.

a(0) = 1; a(n) = Sum_{k=1..n} (4 * k/n - 1) * (k-1)! * binomial(n,k) * a(n-k).

cp's OEIS Frontend

A101612 n! * Sum[k=floor(n/2)..n, 1/k].

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A052767 Expansion of e.g.f.: -(log(1-x))^5.

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A052779 Expansion of e.g.f.: (log(1-x))^6.

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A129841 Antidiagonal sums of triangle T defined in A048594: T(j,k) = k! * Stirling1(j,k), 1<= k <= j.

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A304654 a(n) = (n!)^2 * Sum_{k=1..n-1} 1/(k^2*(n-k)).

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A304655 a(n) = (n!)^3 * Sum_{k=1..n-1} 1/(k^3*(n-k)^2).

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A308346 Expansion of e.g.f. 1/(1 - x)^log(1 - x).

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A333371 Exponential convolution of primorial numbers (A002110) with themselves.

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