A034829 -id:A034829

A053105

a(n) = ((7*n+9)(!^7))/9(!^7), related to A034829 (((7*n+2)(!^7))/2 sept-, or 7-factorials).

+20
3

1, 16, 368, 11040, 408480, 17973120, 916629120, 53164488960, 3455691782400, 248809808332800, 19655974858291200, 1690413837813043200, 157208486916613017600, 15720848691661301760000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Row m=9 of the array A(8; m,n) := ((7*n+m)(!^7))/m(!^7), m >= 0, n >= 0.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..337

FORMULA

a(n) = ((7*n+9)(!^7))/9(!^7)= A034829(n+2)/9.

E.g.f.: 1/(1-7*x)^(16/7).

MATHEMATICA

s=1; lst={s}; Do[s+=n*s; AppendTo[lst, s], {n, 15, 5!, 7}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)

CoefficientList[Series[1/(1-7x)^(16/7), {x, 0, 20}], x]Range[0, 20]! (* Harvey P. Dale, Sep 11 2011 *)

PROG

(PARI) x='x+O('x^30); Vec(serlaplace(1/(1-7*x)^(16/7))) \\ G. C. Greubel, Aug 16 2018

(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-7*x)^(16/7))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 16 2018

CROSSREFS

Cf. A051188, A045754(n+1), A034829-34(n+1), A053104-A053106 (rows m=0..10).

KEYWORD

easy,nonn

AUTHOR

Wolfdieter Lang

STATUS

approved

A034834

One seventh of sept-factorial numbers.

+10
11

1, 14, 294, 8232, 288120, 12101040, 592950960, 33205253760, 2091930986880, 146435169081600, 11275508019283200, 947142673619788800, 86189983299400780800, 8446618363341276518400, 886894928150834034432000, 99332231952893411856384000, 11820535602394316010909696000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..335

Index entries for sequences related to factorial numbers.

FORMULA

7*a(n) = (7*n)(!^7) = Product_{j=1..n} 7*j = 7^n*n!.

E.g.f.: x/(1-7*x).

a(n) = A051188(n)/7.

From Amiram Eldar, Jan 08 2022: (Start)

Sum_{n>=1} 1/a(n) = 7*(exp(1/7)-1).

Sum_{n>=1} (-1)^(n+1)/a(n) = 7*(1-exp(-1/7)). (End)

MATHEMATICA

Table[7^(n-1)*n!, {n, 1, 30}] (* or *) Drop[With[{nn = 50}, CoefficientList[ Series[x/(1-7*x), {x, 0, nn}], x]*Range[0, nn]!], 1] (* G. C. Greubel, Feb 22 2018 *)

PROG

(PARI) my(x='x+O('x^30)); Vec(serlaplace(x/(1-7*x))) \\ G. C. Greubel, Feb 22 2018

(Magma) [7^(n-1)*Factorial(n): n in [1..30]]; // G. C. Greubel, Feb 22 2018

CROSSREFS

Cf. A045754, A034829, A034830, A034831, A034832, A034833, A051188.

KEYWORD

easy,nonn

AUTHOR

Wolfdieter Lang

EXTENSIONS

More terms from G. C. Greubel, Feb 22 2018

STATUS

approved

A034830

a(n) = n-th sept-factorial number divided by 3.

+10
9

1, 10, 170, 4080, 126480, 4806240, 216280800, 11246601600, 663549494400, 43794266630400, 3196981464019200, 255758517121536000, 22250990989573632000, 2091593153019921408000, 211250908455012062208000, 22815098113141302718464000, 2623736283011249812623360000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..339

Index entries for sequences related to factorial numbers.

FORMULA

3*a(n) = (7*n-4)(!^7) = Product_{j=1..n} (7*j-4).

E.g.f.: (-1 + (1-7*x)^(-3/7))/3.

From Amiram Eldar, Dec 20 2022: (Start)

a(n) = A144739(n)/3.

Sum_{n>=1} 1/a(n) = 3*(e/7^4)^(1/7)*(Gamma(3/7) - Gamma(3/7, 1/7)). (End)

MATHEMATICA

Drop[With[{nn = 40}, CoefficientList[Series[(-1 + (1 - 7*x)^(-3/7))/3, {x, 0, nn}], x]*Range[0, nn]!], 1] (* G. C. Greubel, Feb 23 2018 *)

PROG

(PARI) my(x='x+O('x^30)); Vec(serlaplace((-1 + (1-7*x)^(-3/7))/3)) \\ G. C. Greubel, Feb 23 2018

CROSSREFS

Cf. A045754, A034829, A034831, A034832, A034833, A034834, A144739.

KEYWORD

easy,nonn

AUTHOR

Wolfdieter Lang

EXTENSIONS

More terms added by G. C. Greubel, Feb 23 2018

STATUS

approved

A034833

a(n) = n-th sept-factorial number divided by 6.

+10
9

1, 13, 260, 7020, 238680, 9785880, 469722240, 25834723200, 1601752838400, 110520945849600, 8399591884569600, 697166126419276800, 62744951377734912000, 6086260283640286464000, 632971069498589792256000, 70259788714343466940416000, 8290655068292529098969088000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..339

Index entries for sequences related to factorial numbers.

FORMULA

a(n) = A049209(n)/6;

a(n) = (7*n-1)(!^7)/6;

a(n) = (1/6)*Product_{j=1..n} (7*j-1);

a(n) = (1/6)*(7*n)! / (7^n * n! * A045754(n) * 2*A034829(n) * 3*A034830(n) * 4*A034831(n) * 5*A034832(n)).

E.g.f.: (-1 + (1-7*x)^(-6/7))/6.

Sum_{n>=1} 1/a(n) = 6*(e/7)^(1/7)*(Gamma(6/7) - Gamma(6/7, 1/7)). - Amiram Eldar, Dec 20 2022

MATHEMATICA

FoldList[Times, 1, Rest[7*Range[20]-1]] (* Harvey P. Dale, Dec 15 2014 *)

PROG

(PARI) my(x='x+('x^30)); Vec(serlaplace((-1 + (1-7*x)^(-6/7))/6)) \\ G. C. Greubel, Feb 22 2018

(Magma) [(&*[(7*k-1): k in [1..n]])/6: n in [1..30]]; // G. C. Greubel, Feb 24 2018

CROSSREFS

Cf. A049209, A045754, A034829, A034830, A034831, A034832, A034834.

KEYWORD

easy,nonn

AUTHOR

Wolfdieter Lang

STATUS

approved

A034832

a(n) = n-th sept-factorial number divided by 5.

+10
8

1, 12, 228, 5928, 195624, 7824960, 367773120, 19859748480, 1211444657280, 82378236695040, 6178367752128000, 506626155674496000, 45089727855030144000, 4328613874082893824000, 445847229030538063872000, 49043195193359187025920000, 5738053837623024882032640000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..339

Index entries for sequences related to factorial numbers.

FORMULA

5*a(n) = (7*n-2)(!^7) = Product_{j=1..n} (7*j-2).

E.g.f.: (-1 + (1-7*x)^(-5/7))/5.

From Amiram Eldar, Dec 20 2022: (Start)

a(n) = A147585(n+1)/5.

Sum_{n>=1} 1/a(n) = 5*(e/7^2)^(1/7)*(Gamma(5/7) - Gamma(5/7, 1/7)). (End)

MATHEMATICA

Rest[FoldList[Times, 1, 7*Range[20]-2]/5] (* Harvey P. Dale, May 30 2013 *)

Drop[With[{nn = 50}, CoefficientList[Series[(-1 + (1 - 7*x)^(-5/7))/5, {x, 0, nn}], x]*Range[0, nn]!], 1] (* G. C. Greubel, Feb 22 2018 *)

PROG

(PARI) my(x='x+O('x^30)); Vec(serlaplace((-1 + (1-7*x)^(-5/7))/5)) \\ G. C. Greubel, Feb 22 2018

CROSSREFS

Cf. A045754, A034829, A034830, A034831, A034833, A034834, A084947, A147585.

KEYWORD

easy,nonn

AUTHOR

Wolfdieter Lang

EXTENSIONS

More terms from G. C. Greubel, Feb 22 2018

STATUS

approved

A034831

a(n) = n-th sept-factorial number divided by 4.

+10
7

1, 11, 198, 4950, 158400, 6177600, 284169600, 15060988800, 903659328000, 60545174976000, 4480342948224000, 362907778806144000, 31935884534940672000, 3033909030819363840000, 309458721143575111680000, 33731000604649687173120000, 3912796070139363712081920000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..339

Index entries for sequences related to factorial numbers.

FORMULA

4*a(n) = (7*n-3)(!^7) = Product_{j=1..n} (7*j-3).

E.g.f.: (-1 + (1-7*x)^(-4/7))/4.

From Amiram Eldar, Dec 20 2022: (Start)

a(n) = A144827(n)/4.

Sum_{n>=1} 1/a(n) = 4*(e/7^3)^(1/7)*(Gamma(4/7) - Gamma(4/7, 1/7)). (End)

MATHEMATICA

Drop[With[{nn = 40}, CoefficientList[Series[(-1 + (1 - 7*x)^(-4/7))/4, {x, 0, nn}], x]*Range[0, nn]!], 1] (* G. C. Greubel, Feb 22 2018 *)

Table[Product[7 j - 3, {j, n}], {n, 30}]/4 (* Vincenzo Librandi, Feb 24 2018 *)

PROG

(PARI) my(x='x+O('x^30)); Vec(serlaplace((-1 + (1-7*x)^(-4/7))/4)) \\ G. C. Greubel, Feb 22 2018

(Magma) [(&*[(7*k-3): k in [1..n]])/4: n in [1..30]]; // G. C. Greubel, Feb 24 2018

CROSSREFS

Cf. A049209, A045754, A034829, A034830, A034832, A034833, A034834, A144827.

KEYWORD

easy,nonn

AUTHOR

Wolfdieter Lang

EXTENSIONS

More terms added by G. C. Greubel, Feb 23 2018

STATUS

approved

A034835

Expansion of 1/(1-49*x)^(1/7); related to sept-factorial numbers A045754.

+10
6

1, 7, 196, 6860, 264110, 10722866, 450360372, 19365495996, 847240449825, 37560993275575, 1682732498745760, 76028913806967520, 3459315578217022160, 158330213003009860400, 7283189798138453578400, 336483368673996555322080

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..445

A. Straub, V. H. Moll, T. Amdeberhan, The p-adic valuation of k-central binomial coefficients, Acta Arith. 140 (1) (2009) 31-41, eq (1.10)

FORMULA

a(n) = 7^n*A045754(n)/n!, n >= 1, A045754(n) = (7*n-6)(!^7) := product(7*j-6, j=1..n); G.f.: (1-49*x)^(-1/7).

D-finite with recurrence: n*a(n) +7*(-7*n+6)*a(n-1)=0. - R. J. Mathar, Jan 28 2020

MATHEMATICA

CoefficientList[Series[1/(1 - 49*x)^(1/7), {x, 0, 50}], x] (* G. C. Greubel, Feb 22 2018 *)

PROG

(PARI) x='x+O('x^30); Vec(1/(1 - 49*x)^(1/7)) \\ G. C. Greubel, Feb 22 2018

(Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!(1/(1 - 49*x)^(1/7))) // G. C. Greubel, Feb 22 2018

CROSSREFS

Cf. A045754, A004993, A034829, A034830, A034831, A034832, A034833, A034834.

KEYWORD

easy,nonn

AUTHOR

Wolfdieter Lang

STATUS

approved

A053106

a(n) = ((7*n+10)(!^7))/10(1^7), related to A034830 (((7*n+3)(!^7))/3 sept-, or 7-factorials).

+10
5

1, 17, 408, 12648, 480624, 21628080, 1124660160, 66354949440, 4379426663040, 319698146401920, 25575851712153600, 2225099098957363200, 209159315301992140800, 21125090845501206220800

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Row m=10 of the array A(8; m,n) := ((7*n+m)(!^7))/m(!^7), m >= 0, n >= 0.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..337

FORMULA

a(n) = ((7*n+10)(!^7))/10(!^7) = A034830(n+2)/10.

E.g.f.: 1/(1-7*x)^(17/7).

MATHEMATICA

s=1; lst={s}; Do[s+=n*s; AppendTo[lst, s], {n, 16, 5!, 7}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)

With[{nn = 30}, CoefficientList[Series[1/(1 - 7*x)^(17/7), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 16 2018 *)

PROG

(PARI) x='x+O('x^30); Vec(serlaplace(1/(1-7*x)^(17/7))) \\ G. C. Greubel, Aug 16 2018

(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-7*x)^(17/7))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 16 2018

CROSSREFS

Cf. A051188, A045754(n+1), A034829-A034834(n+1), A053104-A053106 (rows m=0..10).

KEYWORD

easy,nonn

AUTHOR

Wolfdieter Lang

STATUS

approved

A053104

a(n) = ((7*n+8)(!^7))/8, related to A045754 ((7*n+1)(!^7) sept-, or 7-factorials).

+10
4

1, 15, 330, 9570, 344520, 14814360, 740718000, 42220926000, 2702139264000, 191851887744000, 14964447244032000, 1271978015742720000, 117021977448330240000, 11585175767384693760000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Row m=8 of the array A(8; m,n) := ((7*n+m)(!^7))/m(!^7), m >= 0, n >= 0.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..337

FORMULA

a(n) = ((7*n+8)(!^7))/8(!^7) = A045754(n+2)/8.

E.g.f.: 1/(1-7*x)^(15/7).

MATHEMATICA

s=1; lst={s}; Do[s+=n*s; AppendTo[lst, s], {n, 14, 5!, 7}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)

With[{nn = 30}, CoefficientList[Series[1/(1 - 7*x)^(15/7), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 16 2018 *)

PROG

(PARI) x='x+O('x^30); Vec(serlaplace(1/(1-7*x)^(15/7))) \\ G. C. Greubel, Aug 16 2018

(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-7*x)^(15/7))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 16 2018

CROSSREFS

Cf. A051188, A045754(n+1), A034829-34(n+1), A053104-A053106 (rows m=0..10).

KEYWORD

easy,nonn

AUTHOR

Wolfdieter Lang

STATUS

approved