A001761 -id:A001761 - cp's OEIS frontend

A357028 E.g.f. satisfies A(x) = (1 - x * A(x))^log(1 - x * A(x)).

1, 0, 2, 6, 82, 820, 13568, 235368, 5111748, 123205248, 3404436312, 103998026880, 3516027852456, 129715202957184, 5198615642907360, 224652658604613120, 10419411912935774736, 516120552745366247424, 27198524267826237745824

Offset: 0

Seiichi Manyama, Sep 09 2022

nonn

Cf. A001761, A357029.

Cf. A357036.

m = 20; (* number of terms *)
A[_] = 0;
Do[A[x_] = (1 - x*A[x])^Log[1 - x*A[x]] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x]*Range[0, m - 1]! (* Jean-François Alcover, Sep 12 2022 *)

a(n) = sum(k=0, n\2, (2*k)!*(n+1)^(k-1)*abs(stirling(n, 2*k, 1))/k!);

E.g.f. satisfies log(A(x)) = log(1 - x * A(x))^2.

a(n) = Sum_{k=0..floor(n/2)} (2*k)! * (n+1)^(k-1) * |Stirling1(n,2*k)|/k!.

A357029 E.g.f. satisfies A(x) = 1/(1 - x * A(x))^(log(1 - x * A(x))^2).

Original entry on oeis.org

1, 0, 0, 6, 36, 210, 3870, 70224, 1122072, 23086344, 586910880, 15469437456, 441107126856, 14206113541152, 496333927370736, 18463733657766144, 739328759822848320, 31759148433997889280, 1447876893211813379520, 69881726567495477445120

Offset: 0

Seiichi Manyama, Sep 09 2022

nonn

Cf. A001761, A357028.

Cf. A353344, A357037.

m = 20; (* number of terms *)
A[_] = 0;
Do[A[x_] = 1/(1 - x*A[x])^(Log[1 - x*A[x]]^2) + O[x]^m // Normal, {m}];
CoefficientList[A[x], x]*Range[0, m - 1]! (* Jean-François Alcover, Sep 12 2022 *)

a(n) = sum(k=0, n\3, (3*k)!*(n+1)^(k-1)*abs(stirling(n, 3*k, 1))/k!);

E.g.f. satisfies log(A(x)) = -log(1 - x * A(x))^3.

a(n) = Sum_{k=0..floor(n/3)} (3*k)! * (n+1)^(k-1) * |Stirling1(n,3*k)|/k!.

A052795 a(n) = (6n)!/(5n+1)!.

Original entry on oeis.org

1, 1, 12, 306, 12144, 657720, 45239040, 3776965920, 371090522880, 41951580652800, 5364506808460800, 765606216965990400, 120639963305775513600, 20803502274492921984000, 3896911902445736638464000, 787971434323820421362688000, 171063718698166603304067072000

Offset: 0

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

Old name was: A simple grammar.

Seiichi Manyama, Table of n, a(n) for n = 0..310
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 752

Cf. A001761, A001763, A365340, A365341.

Cf. A000142, A002295.

spec := [S,{B=Prod(Z,S,S,S,S,S),S=Sequence(B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20); # end of program
seq((6*n)!/(5*n+1)!, n=0..20);  # Mark van Hoeij, May 29 2013

a(n) = (6*n)!/(5*n+1)!; \\ Joerg Arndt, May 29 2013

from sympy import ff
def A052795(n): return ff(6*n,n-1) # Chai Wah Wu, Sep 01 2023

E.g.f.: RootOf(-_Z+_Z^6*x+1).
D-finite Recurrence: {a(1)=1, a(2)=12, (-720-9864*n-48600*n^2-110160*n^3-116640*n^4-46656*n^5)*a(n)+(3125*n^4+9375*n^3+10000*n^2+4500*n+720)*a(n+1), a(6)=45239040, a(3)=306, a(4)=12144, a(5)=657720}.
1/25*3^(1/2)*(5+5^(1/2))^(1/2)*(5-5^(1/2))^(1/2)*Pi^(1/2) *GAMMA(2*n+37/3) *GAMMA(2*n+38/3)/GAMMA(n+34/5)/GAMMA(n+33/5)/GAMMA(n+32/5) /GAMMA(n+36/5) *GAMMA(n+13/2)*3125^(-6-n)*2916^(n+6).
a(n) = (6*n)!/(5*n+1)!. - Mark van Hoeij, May 29 2013
E.g.f.: exp( 1/6 * Sum_{k>=1} binomial(6*k,k) * x^k/k ). - Seiichi Manyama, Feb 08 2024
a(n) = A000142(n)*A002295(n). - Alois P. Heinz, Feb 08 2024
From Seiichi Manyama, Aug 31 2024: (Start)
E.g.f. satisfies A(x) = 1/(1 - x*A(x)^5).
a(n) = Sum_{k=0..n} (5*n+1)^(k-1) * |Stirling1(n,k)|. (End)

New name using Mark van Hoeij's formula from Joerg Arndt, Feb 18 2019

Accidentally removed a(0) reinserted by Georg Fischer, May 09 2021

A105725 Triangle read by rows: T(n,k)=(n+k)!/k! (0<=k<=n-1; n>=1).

Original entry on oeis.org

1, 2, 6, 6, 24, 60, 24, 120, 360, 840, 120, 720, 2520, 6720, 15120, 720, 5040, 20160, 60480, 151200, 332640, 5040, 40320, 181440, 604800, 1663200, 3991680, 8648640, 40320, 362880, 1814400, 6652800, 19958400, 51891840, 121080960, 259459200

Offset: 1

Amarnath Murthy, Apr 18 2005

T(n,n-1)=(2n-1)!/(n-1)! (A000407); T(n,0)=n! (A000142); Row sums yield A092956; Arithmetic means of the rows yield A001761.

Has many diagonals in common with A068424. - Zerinvary Lajos, Apr 14 2006

			1
2 6
6 24 60
24 120 360 840
120 720 2520 6720 15120
720 5040 20160 60480 151200 332640
5040 40320 181440 604800 1663200 3991680 8648640
40320 362880 1814400 6652800 19958400 51891840 121080960 259459200

Cf. A000407, A000142, A092956, A001761.

Maple
```
T:=proc(n,k) if k
				
```

T(n, k)=(n+k)!/k! (0<=k<=n-1; n>=1).

More terms from Emeric Deutsch, Apr 18 2005

A152029 a(n) = 2^n(2n)!/((n+1)!).

Original entry on oeis.org

1, 2, 16, 240, 5376, 161280, 6082560, 276756480, 14760345600, 903333150720, 62412108595200, 4805732361830400, 408117579035443200, 37896632339005440000, 3819980539771748352000, 415422883700177633280000, 48482294191832495554560000, 6044126009248451112468480000

Offset: 0

N. J. A. Sloane, Sep 15 2009

nonn

Robert Israel, Table of n, a(n) for n = 0..334
K. Casteels and B. Stevens, Universal cycles of (n-1)-partitions of an n-set, Discr. Math., 309 (2009), 5332-5340.

[2^n*Factorial(2*n)/Factorial(n+1): n in [0..20]]; // Vincenzo Librandi, Jan 27 2017

seq(2^n*(2*n)!/(n+1)!,n=0..40); # Robert Israel, Jan 25 2017

Table[(2^n) (2 n)! / (n + 1)!, {n, 0, 20}] (* Vincenzo Librandi, Jan 27 2017 *)
With[{nn=20},CoefficientList[Series[2/(1+(1-8x)^(1/2)),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 11 2023 *)

a(n) = 2^n*(2*n)!/(n+1)! \\ Michel Marcus, Jun 02 2013

E.g.f 2/(1+(1-8*x)^(1/2)). - Sergei N. Gladkovskii, Oct 26 2012
a(n) = 2^n * A001761(n) = A065140(n)/(n+1)!. - Michel Marcus, Jun 02 2013
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 2*x/(2*x + (k+2)/((2*k+1)*(2*k+2))/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 03 2013
4*(n+1)*(2*n+1)*a(n) = (n+2)*a(n+1). - Robert Israel, Jan 25 2017
E.g.f.: 1/(1 - 2*x/(1 - 2*x/(1 - 2*x/(1 - 2*x/(1 - 2*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, May 10 2017

A349599 E.g.f. satisfies: log(A(x)) = 1 - exp(-x*A(x)^2).

Original entry on oeis.org

1, 1, 4, 29, 305, 4192, 70875, 1416781, 32551650, 841273527, 24032201213, 747395938962, 24946766300549, 880465276003861, 32274320771151308, 1197240324544640433, 42849289206116498093, 1304855947753532683776, 14954863230501575196551, -2798084168801754024136463

Offset: 0

Seiichi Manyama, Nov 22 2021

sign

a(19) < 0.

Seiichi Manyama, Table of n, a(n) for n = 0..374

Cf. A001761, A001763, A349598.

Cf. A349527, A349602.

a(n) = sum(k=0, n, (-1)^(n-k)*(2*n+1)^(k-1)*stirling(n, k, 2));

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

A357036 E.g.f. satisfies A(x) = (1 - x * A(x))^(log(1 - x * A(x)) / 2).

Original entry on oeis.org

1, 0, 1, 3, 26, 230, 2794, 39564, 663606, 12712104, 275171106, 6632699040, 176309074644, 5123121177096, 161577261004860, 5497133655605760, 200683752698028924, 7825434930630743616, 324616635150708044796, 14273994548639305751040, 663205761925601097418488

Offset: 0

Seiichi Manyama, Sep 09 2022

nonn

Cf. A001761, A357037.

Cf. A347001, A357028.

m = 21; (* number of terms *)
A[_] = 0;
Do[A[x_] = (1 - x*A[x])^(Log[1 - x*A[x]]/2) + O[x]^m // Normal, {m}];
CoefficientList[A[x], x]*Range[0, m - 1]! (* Jean-François Alcover, Sep 12 2022 *)

a(n) = sum(k=0, n\2, (2*k)!*(n+1)^(k-1)*abs(stirling(n, 2*k, 1))/(2^k*k!));

E.g.f. satisfies log(A(x)) = log(1 - x * A(x))^2 / 2.

a(n) = Sum_{k=0..floor(n/2)} (2*k)! * (n+1)^(k-1) * |Stirling1(n,2*k)|/(2^k * k!).

A357037 E.g.f. satisfies A(x) = 1/(1 - x * A(x))^(log(1 - x * A(x))^2 / 6).

Original entry on oeis.org

1, 0, 0, 1, 6, 35, 295, 3304, 42112, 599724, 9657330, 174222576, 3464835726, 75208002792, 1771121398956, 44998593873024, 1226723273550720, 35714547582173280, 1106012915718532920, 36304411160854523520, 1259105580819317636280, 46007354360033491345920

Offset: 0

Seiichi Manyama, Sep 09 2022

nonn

Cf. A001761, A357036.

Cf. A347002, A357029, A357032.

m = 22; (* number of terms *)
A[_] = 0;
Do[A[x_] = 1/(1 - x*A[x])^(Log[1 - x*A[x]]^2/6) + O[x]^m // Normal, {m}];
CoefficientList[A[x], x]*Range[0, m - 1]! (* Jean-François Alcover, Sep 12 2022 *)

a(n) = sum(k=0, n\3, (3*k)!*(n+1)^(k-1)*abs(stirling(n, 3*k, 1))/(6^k*k!));

E.g.f. satisfies log(A(x)) = -log(1 - x * A(x))^3 / 6.

a(n) = Sum_{k=0..floor(n/3)} (3*k)! * (n+1)^(k-1) * |Stirling1(n,3*k)|/(6^k * k!).

A355767 E.g.f. satisfies A(x)^(A(x)^2) = 1/(1 - x*A(x)).

Original entry on oeis.org

1, 1, 0, 6, -4, 300, -828, 42224, -266992, 11916864, -132472320, 5688511488, -95465876064, 4138883728512, -95019458907072, 4276023328128000, -125481256750340352, 5958015717717504000, -212934915549001078272, 10767675634298110255104

Offset: 0

Seiichi Manyama, Jul 16 2022

sign

Cf. A001761, A141209.

Cf. A355768.

a(n) = sum(k=0, n, (n-2*k+1)^(k-1)*abs(stirling(n, k, 1)));

a(n) = Sum_{k=0..n} (n-2*k+1)^(k-1) * |Stirling1(n,k)|.

A064307 Triangle of coefficients of certain numerator polynomials N(n,x).

Original entry on oeis.org

1, 1, 0, 1, 2, 1, 1, 10, 17, 2, 1, 37, 181, 111, 6, 1, 126, 1530, 2624, 741, 18, 1, 422, 11607, 43940, 34063, 4950, 57, 1, 1422, 83823, 616894, 1013799, 412698, 33337, 186, 1, 4853, 593203, 7846573, 23794925

Offset: 1

Wolfdieter Lang, Sep 13 2001

The g.f. for the sequence in the subdiagonal d>=1 (main diagonal: d=0) of triangle A064094 is N(d,x)/(1-x)^d.

Row sums give A001761(n+1). Main diagonal gives A000957(n+1), n >= 0.

			Triangle begins:
  1;
  1, 0;
  1, 2,  1;       N(3,x) = 1+2*x+x^2 = (1+x)^2.
  1, 10, 17,  2;
  1, 37, 181, 111, 6;
  ...

Cf. A000957, A001761, A064094.

a(n, m) = [x^m]N(n, x); N(n, x) = (1-x)^(n-1) + Sum_{k=1..n-1} A064308(n-1, k)*k!*x^k*(1-x)^(n-1-k) for n >= 2; N(1, x) = 1 = N(2, x).

cp's OEIS Frontend

A357028 E.g.f. satisfies A(x) = (1 - x * A(x))^log(1 - x * A(x)).

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A357029 E.g.f. satisfies A(x) = 1/(1 - x * A(x))^(log(1 - x * A(x))^2).

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A052795 a(n) = (6*n)!/(5*n+1)!.

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A105725 Triangle read by rows: T(n,k)=(n+k)!/k! (0<=k<=n-1; n>=1).

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A152029 a(n) = 2^n*(2*n)!/((n+1)!).

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A349599 E.g.f. satisfies: log(A(x)) = 1 - exp(-x*A(x)^2).

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A357036 E.g.f. satisfies A(x) = (1 - x * A(x))^(log(1 - x * A(x)) / 2).

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A357037 E.g.f. satisfies A(x) = 1/(1 - x * A(x))^(log(1 - x * A(x))^2 / 6).

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A052795 a(n) = (6n)!/(5n+1)!.

A152029 a(n) = 2^n(2n)!/((n+1)!).