A007696 -id:A007696 - cp's OEIS frontend

A004130 Numerators in expansion of (1-x)^{-1/4}.

1, 1, 5, 15, 195, 663, 4641, 16575, 480675, 1762475, 13042315, 48612265, 729183975, 2748462675, 20809788825, 79077197535, 4823709049635, 18443593425075, 141400882925575, 543277076503525, 8366466978154285, 32270658344309385

Offset: 0

N. J. A. Sloane

Numerators in expansion of sqrt(1/sqrt(1-4x)). - Paul Barry, Jul 12 2005

Denominators are in A088802. - Michael Somos, Aug 23 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A004134, A004981, A034255, A034385, A048882, A007696, A000265, A049606.

Table[Numerator[Binomial[-1/4, n] (-1)^n], {n, 0, 20}]

{a(n) = if( n<0, 0, numerator( polcoeff( (1 - x +x*O(x^n))^(-1/4), n ) ) ) } /* Michael Somos, Aug 23 2007 */

a(n) = prod(k=1, n, (4k-3)/k * 2^A007814(k)), proved by Mitch Harris, following a conjecture by Ralf Stephan.
a(n) = 2^(e_2((2n)!)-n)/n! Product[4k+1,{k,0,n-1}], where e_2((2n)!) is the highest power of 2 that divides (2n)! (sequence A005187). - Emanuele Munarini, Jan 25 2011
Numerators in (1-4t)^(-1/4) = 1 + t + (5/2)t^2 + (15/2)t^3 + (195/8)t^4 + (663/8)t^5 + (4641/16)t^6 + (16575/16)t^7 + ... = 1 + t + 5*t^2/2! + 45*t^3/3! + 585*t^4/4! + ... = e.g.f. for the quartic factorials A007696 (cf. A094638). - Tom Copeland, Dec 04 2013

A034975 One seventh of octo-factorial numbers.

Original entry on oeis.org

1, 15, 345, 10695, 417105, 19603935, 1078216425, 67927634775, 4822862069025, 381006103452975, 33147531000408825, 3149015445038838375, 324348590839000352625, 36002693583129039141375, 4284320536392355657823625, 544108708121829168543600375, 73454675596446937753386050625

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..325
Index entries for sequences related to factorial numbers.

Cf. A007696, A034176, A034908, A034909, A034910, A034911, A034912, A034976, A045755, A049210.

[n le 1 select 1 else (8*n-1)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 21 2022

Table[8^n*Pochhammer[7/8, n]/7, {n, 40}] (* G. C. Greubel, Oct 21 2022 *)

[8^n*rising_factorial(7/8,n)/7 for n in range(1,40)] # G. C. Greubel, Oct 21 2022

7*a(n) = (8*n-1)!^8 = Product_{j=1..n} (8*j-1) = (8*n)!/((2*n)!*2^(6*n)*3^2*5 * A045755(n)*A007696(n)*A034909(n)*A034911(n)*A034176(n)).
E.g.f.: (-1+(1-8*x)^(-7/8))/7.
G.f.: x/(1-15*x/(1-8*x/(1-23*x/(1-16*x/(1-31*x/(1-24*x/(1-39*x/(1-32*x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012
a(n) = (1/7) * 8^n * Pochhammer(n, 7/8). - G. C. Greubel, Oct 21 2022
From Amiram Eldar, Dec 20 2022: (Start)
a(n) = A049210(n)/7.
Sum_{n>=1} 1/a(n) = 7*(e/8)^(1/8)*(Gamma(7/8) - Gamma(7/8, 1/8)). (End)

A113131 a(0) = a(1) = 1, a(2) = x, a(3) = 2x^2, a(n) = x(n-1)a(n-1) + Sum_{j=2..n-2} (j-1)a(j)a(n-j), n>=4 and for x = 4.

Original entry on oeis.org

1, 1, 4, 32, 400, 6784, 144128, 3658752, 107686656, 3599697920, 134617038848, 5567255822336, 252278661832704, 12431395516383232, 661885541595873280, 37869659304097218560, 2317293119684500193280, 151022143036329696952320

Offset: 0

Philippe Deléham and Paul D. Hanna, Oct 28 2005

nonn

			a(2) = 4.
a(3) = 2*4^2 = 32.
a(4) = 4*3*32 + 1*4*4 = 400.
a(5) = 4*4*400 + 1*4*32 + 2*32*4 = 6784.
a(6) = 4*5*6784 + 1*4*400 + 2*32*32 + 3*400*4 = 144128.
G.f.: A(x) = 1 + x + 4*x^2 + 32*x^3 + 400*x^4 + 6784*x^5 +...
= x/series_reversion(x + x^2 + 5*x^3 + 45*x^4 + 585*x^5 +...).

Cf. A007696, A075834(x=1), A111088(x=2), A113130(x=3), A113132(x=5), A113133(x=6), A113134(x=7), A113135(x=8).

x=4;a[0]=a[1]=1;a[2]=x;a[3]=2x^2;a[n_]:=a[n]=x*(n-1)*a[n-1]+Sum[(j-1)*a[j ]*a[n-j], {j, 2, n-2}];Table[a[n], {n, 0, 18}](Robert G. Wilson v)

a(n)=Vec(x/serreverse(x*Ser(vector(n+1,k,if(k==1,1, prod(j=0,k-2,4*j+1))))))[n+1]

a(n,x=4)=if(n<0,0,if(n==0 || n==1,1,if(n==2,x,if(n==3,2*x^2,x*(n-1)*a(n-1)+sum(j=2,n-2,(j-1)*a(j)*a(n-j))))))

a(n+1) = Sum{k, 0<=k<=n} 4^k*A113129(n, k).
G.f.: A(x) = x/series_reversion(x*G(x)) where G(x) = g.f. of quartic factorials (A007696).
G.f. satisfies: A(x*G(x)) = G(x) = g.f. of triple factorials (A007696).

A264781 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive pattern 45321; triangle T(n,k), n >= 0, 0 <= k <= max(0, floor((n-1)/4)), read by rows.

Original entry on oeis.org

1, 1, 2, 6, 24, 119, 1, 708, 12, 4914, 126, 38976, 1344, 347765, 15110, 5, 3447712, 180736, 352, 37598286, 2308548, 9966, 447294144, 31481472, 225984, 5764747515, 457520055, 4753185, 45, 80011430240, 7068885600, 97954080, 21280, 1189835682714, 115808906178

Offset: 0

Alois P. Heinz, Nov 24 2015

Consecutive patterns 12354, 21345, 54312 give the same triangle.

The attached Maple program gives a recurrence for the o.g.f. of each row in terms of u. Using that recurrence we may get any row or column from this irregular triangular array T(n,k). The recurrence follows from manipulation of the bivariate o.g.f./e.g.f. 1/W(u,z) = Sum_{n, k >= 0} T(n, k)*u^k*z^n/n!, whose reciprocal W(u,z) is the solution of the o.d.e. in Theorem 3.2 in Elizalde and Noy (2003) (with m = a = 3). - Petros Hadjicostas, Nov 05 2019

			T(5,1) = 1: 45321.
T(6,1) = 12: 156432, 256431, 356421, 453216, 456321, 463215, 546321, 563214, 564213, 564312, 564321, 645321.
T(9,2) = 5: 786549321, 796548321, 896547321, 897546321, 897645321.
Triangle T(n,k) begins:
00 :           1;
01 :           1;
02 :           2;
03 :           6;
04 :          24;
05 :         119,          1;
06 :         708,         12;
07 :        4914,        126;
08 :       38976,       1344;
09 :      347765,      15110,        5;
10 :     3447712,     180736,      352;
11 :    37598286,    2308548,     9966;
12 :   447294144,   31481472,   225984;
13 :  5764747515,  457520055,  4753185,    45;
14 : 80011430240, 7068885600, 97954080, 21280;

Alois P. Heinz, Rows n = 0..170, flattened
A. Baxter, B. Nakamura, and D. Zeilberger, Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes, 2011.
Sergi Elizalde and Marc Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-125; see Theorem 3.2 (p. 116) with m = a = 3.
Petros Hadjicostas, Maple program for a recurrence.

Columns k=0-1 give: A202213, A264896.
Row sums give A000142.
T(4n+1,n) gives A007696.
Cf. A002265, A007696, A062199.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1, add(
       b(u+j-1, o-j, `if`(u+j-30, -1, `if`(t=-1, -2, 0)))), j=1..u)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..17);

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Sum[
     b[u+j-1, o-j, If[u+j-3 < j, 0, j]], {j, 1, o}] + Expand[
     If[t == -2, x, 1]*Sum[b[u-j, o+j-1, If[j < t || t == -2, 0,
     If[t > 0, -1, If[t == -1, -2, 0]]]], {j, 1, u}]]];
T[n_] := CoefficientList[b[n, 0, 0], x];
T /@ Range[0, 17] // Flatten (* Jean-François Alcover, Feb 19 2021, after Alois P. Heinz *)

Sum_{k > 0} k * T(n,k) = A062199(n-5) for n > 4.

A347016 Expansion of e.g.f. 1 / (1 + 4 * log(1 - x))^(1/4).

Original entry on oeis.org

1, 1, 6, 62, 916, 17644, 419360, 11859840, 388965600, 14514046560, 607165485120, 28143329181120, 1431690475207680, 79302863940387840, 4751108622148907520, 306118435580577146880, 21107196651940518551040, 1550773243761690603179520, 120947288498720390755353600

Offset: 0

Ilya Gutkovskiy, Aug 10 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..360

Cf. A007696, A007840, A346978, A346983, A347015.

g:= proc(n) option remember; `if`(n<2, 1, (4*n-3)*g(n-1)) end:
a:= n-> add(abs(Stirling1(n, k))*g(k), k=0..n):
seq(a(n), n=0..18);  # Alois P. Heinz, Aug 10 2021

nmax = 18; CoefficientList[Series[1/(1 + 4 Log[1 - x])^(1/4), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] 4^k Pochhammer[1/4, k], {k, 0, n}], {n, 0, 18}]

a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A007696(k).
a(n) ~ n! * exp(n/4) / (Gamma(1/4) * 2^(1/2) * n^(3/4) * (exp(1/4) - 1)^(n + 1/4)). - Vaclav Kotesovec, Aug 14 2021
a(0) = 1; a(n) = Sum_{k=1..n} (4 - 3*k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023

A352073 Expansion of e.g.f. 1/(1 - log(1 + 4*x))^(1/4).

Original entry on oeis.org

1, 1, 1, 17, 1, 1889, -12415, 631665, -11224575, 461864385, -13754112255, 596055636945, -24148300842495, 1181210529292065, -59009709972278655, 3297137505670374705, -193318225258785780735, 12263541239089421903745, -820804950905249837195775

Offset: 0

Seiichi Manyama, Mar 05 2022

sign

Cf. A006252, A097397, A352070.

Cf. A007696, A008545, A352119.

m = 18; Range[0, m]! * CoefficientList[Series[(1 - Log[1 + 4*x])^(-1/4), {x, 0, m}], x] (* Amiram Eldar, Mar 05 2022 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+4*x))^(1/4)))

a(n) = sum(k=0, n, 4^(n-k)*prod(j=0, k-1, 4*j+1)*stirling(n, k, 1));

a(n) = Sum_{k=0..n} 4^(n-k) * (Product_{j=0..k-1} (4*j+1)) * Stirling1(n,k).

a(0) = 1; a(n) = Sum_{k=1..n} (-4)^k * (3/4 * k/n - 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 18 2023

A051620 a(n) = (4n+8)(!^4)/8(!^4), related to A034177(n+1) ((4n+4)(!^4) quartic, or 4-factorials).

Original entry on oeis.org

1, 12, 192, 3840, 92160, 2580480, 82575360, 2972712960, 118908518400, 5231974809600, 251134790860800, 13059009124761600, 731304510986649600, 43878270659198976000, 2808209322188734464000, 190958233908833943552000

Offset: 0

Wolfdieter Lang

Row m=8 of the array A(5; m,n) := ((4*n+m)(!^4))/m(!^4), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..363

Cf. A047053, A007696(n+1), A000407, A034176(n+1), A034177(n+1), A051617-A051622 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-4*x)^(12/4))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

G(x):=(1-4*x)^(n-4): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od:x:=0:seq(f[n],n=0..15); # Zerinvary Lajos, Apr 04 2009

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 11, 5!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn=20},CoefficientList[Series[1/(1-4*x)^3,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 10 2017 *)

x='x+O('x^30); Vec(serlaplace(1/(1-4*x)^(12/4))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((4*n+8)(!^4))/8(!^4) = A034177(n+2)/8.
E.g.f.: 1/(1-4*x)^3.
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 2*x/(2*x + 1/(2*k+6)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 02 2013

A134275 Triangle of numbers obtained from the partition array A134274.

Original entry on oeis.org

1, 5, 1, 45, 5, 1, 585, 70, 5, 1, 9945, 810, 70, 5, 1, 208845, 14895, 935, 70, 5, 1, 5221125, 284895, 16020, 935, 70, 5, 1, 151412625, 7055100, 309645, 16645, 935, 70, 5, 1, 4996616625, 192734100, 7526475, 315270, 16645, 935, 70, 5, 1, 184874815125

Offset: 1

Wolfdieter Lang, Nov 13 2007

This triangle is named S2(5)'.

In the same manner the unsigned Lah triangle A008297 is obtained from the partition array A130561.

			Triangle begins:
  [1];
  [5,1];
  [45,5,1];
  [585,70,5,1];
  [9945,810,70,5,1];
  ...

Wolfdieter Lang, First 10 rows and more.

Cf. A134276 (row sums). A134277 (alternating row sums).

Cf. A134151 (S2(4)').

a(n,m) = sum(product(S2(5;j,1)^e(n,m,q,j),j=1..n),q=1..p(n,m)) if n>=m>=1, else 0. Here p(n,m)=A008284(n,m), the number of m parts partitions of n and e(n,m,q,j) is the exponent of j in the q-th m part partition of n. S2(5;j,1)= A007696(j) = A049029(j,1) = (4*j-3)(!^4), (quadruple- or 4-factorials).

A024382 a(n) = n-th elementary symmetric function of the first n+1 positive integers congruent to 1 mod 4.

Original entry on oeis.org

1, 6, 59, 812, 14389, 312114, 8011695, 237560280, 7990901865, 300659985630, 12511934225955, 570616907588100, 28301322505722525, 1516683700464669450, 87336792132539066775, 5378036128829898836400, 352652348707389385916625, 24533212082483855129037750

Offset: 0

Clark Kimberling

nonn

a(n) is equal to the determinant of the n X n matrix whose (i,j)-entry is KroneckerDelta[i,j]((4*i+2)-1)+1. - John M. Campbell, May 23 2011
From R. J. Mathar, Oct 01 2016: (Start)
The k-th elementary symmetric functions of the integers 1+4*j, j=1..n, form a triangle T(n,k), 0<=k<=n, n>=0:
1
1 1
1 6 5
1 15 59 45
1 28 254 812 585
1 45 730 5130 14389 9945
1 66 1675 20460 122119 312114 208845
1 91 3325 62335 633619 3365089 8011695 5221125
1 120 5964 158760 2441334 21740040 105599276 237560280 151412625
This here is the first subdiagonal. The diagonal seems to be A007696. The 2nd column is A000384, the 3rd A024378, the 4th A024379. (End)

			For n = 1 we have a(1) = 1*5*(1/1 + 1/5) = 6.
For n = 2 we have a(2) = 1*5*9*(1/1 + 1/5 + 1/9) = 59.
For n = 3 we have a(3) = 1*5*9*13*(1/1 + 1/5 + 1/9 + 1/13) = 812. - _Gheorghe Coserea_, Dec 24 2015

Alois P. Heinz, Table of n, a(n) for n = 0..350

Cf. A024216.

I:=[1,6]; [n le 2 select I[n] else (8*n-10)*Self(n-1)-(4*n-7)^2*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 26 2015

a:= proc(n) option remember; `if`(n<3, [1, 6, 59][n+1],
      (8*n-2)*a(n-1) -(4*n-3)^2*a(n-2))
    end;
seq(a(n), n=0..20);  # Alois P. Heinz, Feb 25 2015

Table[Det[Array[KroneckerDelta[#1,#2]((4*#1+2)-1)+1&,{k, k}]],{k,1,10}] (* John M. Campbell, May 23 2011 *)
RecurrenceTable[{a[0] == 1, a[1] == 6, a[n] == (8 n - 2) a[n - 1] - (4 n - 3)^2 a[n - 2]}, a, {n, 0, 20}] (* Vincenzo Librandi, Dec 26 2015 *)

x = 'x + O('x^33); Vec(serlaplace((4-log(1-4*x))/(4*(1-4*x)^(5/4)))) \\ Gheorghe Coserea, Dec 24 2015

a(n) = (8*n-2)*a(n-1) - (4*n-3)^2*a(n-2) for n>1. - Alois P. Heinz, Feb 25 2015

E.g.f.: (4-log(1-4*x))/(4*(1-4*x)^(5/4)). - Gheorghe Coserea, Dec 24 2015

More terms from Alois P. Heinz, Feb 25 2015

A051619 a(n) = (4n+7)(!^4)/7(!^4), related to A034176(n+1) ((4n+3)(!^4) quartic, or 4-factorials).

Original entry on oeis.org

1, 11, 165, 3135, 72105, 1946835, 60351885, 2112315975, 82380323025, 3542353890075, 166490632833525, 8491022274509775, 467006225098037625, 27553367280784219875, 1735862138689405852125, 116302763292190192092375

Offset: 0

Wolfdieter Lang

Row m=7 of the array A(5; m,n) := ((4*n+m)(!^4))/m(!^4), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..363

Cf. A047053, A007696(n+1), A000407, A034176(n+1), A034177(n+1), A051617-A051622 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-4*x)^(11/4))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 10, 5!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 4*x)^(11/4), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-4*x)^(11/4))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((4*n+7)(!^4))/7(!^4) = A034176(n+2)/7.

E.g.f.: 1/(1-4*x)^(11/4).

cp's OEIS Frontend

A004130 Numerators in expansion of (1-x)^{-1/4}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A034975 One seventh of octo-factorial numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A113131 a(0) = a(1) = 1, a(2) = x, a(3) = 2x^2, a(n) = x*(n-1)*a(n-1) + Sum_{j=2..n-2} (j-1)*a(j)*a(n-j), n>=4 and for x = 4.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A264781 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive pattern 45321; triangle T(n,k), n >= 0, 0 <= k <= max(0, floor((n-1)/4)), read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A347016 Expansion of e.g.f. 1 / (1 + 4 * log(1 - x))^(1/4).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A352073 Expansion of e.g.f. 1/(1 - log(1 + 4*x))^(1/4).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A051620 a(n) = (4*n+8)(!^4)/8(!^4), related to A034177(n+1) ((4*n+4)(!^4) quartic, or 4-factorials).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A134275 Triangle of numbers obtained from the partition array A134274.

A113131 a(0) = a(1) = 1, a(2) = x, a(3) = 2x^2, a(n) = x(n-1)a(n-1) + Sum_{j=2..n-2} (j-1)a(j)a(n-j), n>=4 and for x = 4.

A051620 a(n) = (4n+8)(!^4)/8(!^4), related to A034177(n+1) ((4n+4)(!^4) quartic, or 4-factorials).

A051619 a(n) = (4n+7)(!^4)/7(!^4), related to A034176(n+1) ((4n+3)(!^4) quartic, or 4-factorials).