cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A104671 a(n) = binomial(n+3,n)*binomial(n+8,n).

Original entry on oeis.org

1, 36, 450, 3300, 17325, 72072, 252252, 772200, 2123550, 5348200, 12514788, 27511848, 57316350, 113954400, 217443600, 400096224, 712671399, 1232995500, 2077825750, 3418915500, 5504453955, 8687301480, 13461727500, 20510685000, 30766027500, 45484495056
Offset: 0

Views

Author

Zerinvary Lajos, Apr 22 2005

Keywords

Examples

			If n=0 then C(0+3,3)*C(0+8,0+0) = C(3,3)*C(8,0) = 1*1 = 1.
If n=6 then C(6+3,3)*C(6+8,6+0) = C(9,3)*C(14,6) = 84*3003 = 252252.

Links

Crossrefs

Cf. A062190.

Programs

  • Magma
    A104671:= func< n | Binomial(n+3,n)*Binomial(n+8,n) >;
[A104671(n): n in [0..30]]; // G. C. Greubel, Mar 01 2025
  • Mathematica
    Table[Binomial[n+3,3]Binomial[n+8,n],{n,0,30}] (* or *)
LinearRecurrence[ {12,-66,220,-495,792,-924,792,-495,220,-66,12,-1},{1,36,450,3300,17325,72072,252252,772200,2123550,5348200,12514788,27511848},30] (* Harvey P. Dale, Oct 05 2017 *)
  • SageMath
    def A104671(n): return binomial(n+3,n)*binomial(n+8,n)
print([A104671(n) for n in range(31)]) # G. C. Greubel, Mar 01 2025

Formula

G.f.: (1+24*x+84*x^2+56*x^3)/(1-x)^12. - Bruno Berselli, Jun 06 2012
From Amiram Eldar, Aug 30 2022: (Start)
Sum_{n>=0} 1/a(n) = 144*Pi^2 - 1739736/1225.
Sum_{n>=0} (-1)^n/a(n) = 16*Pi^2 - 13312*log(2)/35 - 515202/1225. (End)

Extensions

a(12) corrected by Colin Barker, Jun 06 2012
More terms and a(7), a(15) corrected by Bruno Berselli, Jun 06 2012

A104672 a(n) = binomial(n+4,n)*binomial(n+9,n).

Original entry on oeis.org

1, 50, 825, 7700, 50050, 252252, 1051050, 3775200, 12033450, 34763300, 92470378, 229265400, 534952600, 1183859600, 2500601400, 5067885504, 9898213875, 18700431750, 34284124875, 61160599500, 106419443130, 180985447500, 301393121250, 492256440000, 789661372500
Offset: 0

Views

Author

Zerinvary Lajos, Apr 22 2005

Keywords

Examples

			If n=0 then C(0+4,4)*C(0+9,0+0) = C(4,4)*C(9,0) = 1*1 = 1.
If n=6 then C(6+4,4)*C(6+9,6+0) = C(10,4)*C(15,6) = 210*5005 = 1051050.

Links

Crossrefs

Cf. A062190.

Programs

  • Magma
    A104672:= func< n | Binomial(n+4,n)*Binomial(n+9,n) >;
[A104672(n): n in [0..30]]; // G. C. Greubel, Mar 01 2025
  • Mathematica
    Table[Binomial[n+4,4]Binomial[n+9,n],{n,0,20}] (* Harvey P. Dale, Nov 15 2018 *)
  • SageMath
    def A104672(n): return binomial(n+4,n)*binomial(n+9,n)
print([A104672(n) for n in range(31)]) # G. C. Greubel, Mar 01 2025

Formula

From Amiram Eldar, Aug 30 2022: (Start)
Sum_{n>=0} 1/a(n) = 990*Pi^2 - 38297957/3920.
Sum_{n>=0} (-1)^n/a(n) = 15*Pi^2 - 12288*log(2)/7 + 4193253/3920. (End)
G.f.: (1 + 36*x + 216*x^2 + 336*x^3 + 126*x^4)/(1-x)^14. - G. C. Greubel, Mar 01 2025

Extensions

More terms from Harvey P. Dale, Nov 15 2018

A104674 a(n) = binomial(n+6, 6) * binomial(n+11, n).

Original entry on oeis.org

1, 84, 2184, 30576, 286650, 2018016, 11435424, 54609984, 226972746, 840639800, 2824549728, 8730426432, 25099975992, 67725379008, 172768824000, 419252346240, 972796459635, 2167754048460, 4656656844840, 9674494830000, 19494107082450, 38192536324800, 72913023892800
Offset: 0

Views

Author

Zerinvary Lajos, Apr 22 2005

Keywords

Examples

			If n=0 then C(0+6,6)*C(0+11,0+0) = C(6,6)*C(11,0) = 1*1 = 1.
If n=8 then C(8+6,6)*C(8+11,8+0) = C(14,6)*C(19,8) = 3003*75582 = 226972746.

Links

Crossrefs

Cf. A062190.

Programs

  • Magma
    A104674:= func< n | Binomial(n+6,n)*Binomial(n+11,n) >;
[A104674(n): n in [0..30]]; // G. C. Greubel, Mar 01 2025
  • Mathematica
    a[n_] := Binomial[n + 6, 6] * Binomial[n + 11, n]; Array[a, 25, 0] (* Amiram Eldar, Aug 30 2022 *)
  • PARI
    a(n)={binomial(n+6, 6) * binomial(n+11, n)} \\ Andrew Howroyd, Nov 08 2019
  • SageMath
    def A104674(n): return binomial(n+6,n)*binomial(n+11,n)
print([A104674(n) for n in range(31)]) # G. C. Greubel, Mar 01 2025

Formula

From Amiram Eldar, Aug 30 2022: (Start)
Sum_{n>=0} 1/a(n) = 33033*Pi^2 - 16431490141/50400.
Sum_{n>=0} (-1)^n/a(n) = 1171456*log(2)/105 - 11*Pi^2/2 - 1625274871/211680. (End)
G.f.: (1 + 66*x + 825*x^2 + 3300*x^3 + 4950*x^4 + 2772*x^5 + 462*x^6)/(1 - x)^18. - G. C. Greubel, Mar 01 2025

Extensions

a(8) corrected and terms a(9) and beyond from Andrew Howroyd, Nov 08 2019

A104675 a(n) = C(n+1,n) * C(n+6,1).

Original entry on oeis.org

6, 14, 24, 36, 50, 66, 84, 104, 126, 150, 176, 204, 234, 266, 300, 336, 374, 414, 456, 500, 546, 594, 644, 696, 750, 806, 864, 924, 986, 1050, 1116, 1184, 1254, 1326, 1400, 1476, 1554, 1634, 1716, 1800, 1886, 1974, 2064, 2156, 2250, 2346, 2444, 2544, 2646
Offset: 0

Views

Author

Zerinvary Lajos, Apr 22 2005

Keywords

Examples

			If n=0 then C(0+1,0+0) * C(0+6,1) = C(1,0) * C(6,1) = 1*6 = 6.
If n=5 then C(5+1,5+0) * C(5+6,1) = C(6,5) * C(11,1) = 6*11 = 66.

Links

Crossrefs

Cf. A002522, A028557, A062190.

Programs

  • Magma
    [(n+1)*(n+6): n in [0..50]]; // G. C. Greubel, Mar 01 2025
  • Mathematica
    Table[Binomial[n + 1, n] Binomial[n + 6, 1], {n, 0, 48}] (* or *)
CoefficientList[Series[2 (3 - 2 x)/(1 - x)^3, {x, 0, 49}], x] (* or *)
LinearRecurrence[{3, -3, 1}, {6, 14, 24}, 49] (* Michael De Vlieger, Apr 06 2017 *)
  • PARI
    Vec(2*(3 - 2*x) / (1 - x)^3 + O(x^80)) \\ Colin Barker, Apr 06 2017
  • PARI
    a(n)=(n+6)*(n+1) \\ Charles R Greathouse IV, Jun 17 2017
  • Python
    from sympy import binomial
def a(n): return binomial(n + 1, n) * binomial(n + 6, 1) # Indranil Ghosh, Apr 06 2017

Formula

a(n) = (n+1)*(n+6) = A028557(n+1). - R. J. Mathar, May 19 2008
a(n) = 2*n + a(n-1) + 6 (with a(0)=6). Vincenzo Librandi, Nov 13 2010
From Colin Barker, Apr 06 2017: (Start)
G.f.: 2*(3 - 2*x) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. (End)
E.g.f.: exp(x)*(x^2 + 8x + 6). - Indranil Ghosh, Apr 06 2017
From Amiram Eldar, Aug 30 2022: (Start)
Sum_{n>=0} 1/a(n) = 137/300.
Sum_{n>=0} (-1)^n/a(n) = 2*log(2)/5 - 47/300. (End)

A104676 a(n) = binomial(n+2,2) * binomial(n+7,2).

Original entry on oeis.org

21, 84, 216, 450, 825, 1386, 2184, 3276, 4725, 6600, 8976, 11934, 15561, 19950, 25200, 31416, 38709, 47196, 57000, 68250, 81081, 95634, 112056, 130500, 151125, 174096, 199584, 227766, 258825, 292950, 330336, 371184, 415701, 464100, 516600, 573426, 634809, 700986
Offset: 0

Views

Author

Zerinvary Lajos, Apr 22 2005

Keywords

Examples

			If n=0 then C(0+2,0+0)*C(0+7,2) = C(2,0)*C(7,2) = 1*21 = 21.
If n=8 then C(8+2,8+0)*C(8+7,2) = C(10,8)*C(15,2) = 45*105 = 4725.

Links

Crossrefs

Cf. A000217, A062190.
Subsequence of A085780.

Programs

Formula

From R. J. Mathar, Nov 29 2015: (Start)
a(n) = A000217(n+1) * A000217(n+6).
G.f.: 3*(7 - 7*x + 2*x^2)/(1-x)^5. (End)
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - Wesley Ivan Hurt, Jan 25 2022
From Amiram Eldar, Aug 30 2022: (Start)
Sum_{n>=0} 1/a(n) = 7/100.
Sum_{n>=0} (-1)^n/a(n) = 7/180. (End)
E.g.f.: (1/4)*(84 + 252*x + 138*x^2 + 22*x^3 + x^4)*exp(x). - G. C. Greubel, Mar 01 2025

A104677 a(n) = binomial(n+3,3)*binomial(n+8,3).

Original entry on oeis.org

56, 336, 1200, 3300, 7700, 16016, 30576, 54600, 92400, 149600, 233376, 352716, 518700, 744800, 1047200, 1445136, 1961256, 2622000, 3458000, 4504500, 5801796, 7395696, 9338000, 11687000, 14508000, 17873856, 21865536, 26572700, 32094300, 38539200, 46026816
Offset: 0

Views

Author

Zerinvary Lajos, Apr 22 2005

Keywords

Examples

			If n=0 then C(0+3,0+0)*C(0+8,3) = C(3,0)*C(8,3) = 1*56 = 56.
If n=8 then C(8+3,8+0)*C(8+8,3) = C(11,8)*C(16,3) = 165*560 = 92400.

Links

Crossrefs

Cf. A000292, A033276, A062190.

Programs

  • Mathematica
    a[n_] :=  Binomial[n+3, 3] * Binomial[n+8, 3]; Array[a, 30, 0] (* Amiram Eldar, Aug 30 2022 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{56,336,1200,3300,7700,16016,30576},40] (* Harvey P. Dale, Jan 06 2023 *)

Formula

From R. J. Mathar, Nov 29 2015: (Start)
a(n) = A000292(n+1)*A000292(n+6) = 4*A033276(n+6).
G.f.: 4*(-14+14*x-6*x^2+x^3)/(x-1)^7. (End)
From Amiram Eldar, Aug 30 2022: (Start)
Sum_{n>=0} 1/a(n) = 109/4900.
Sum_{n>=0} (-1)^n/a(n) = 48*log(2)/35 - 2291/2450. (End)

A104679 a(n) = C(n+5,5)*C(n+10,5).

Original entry on oeis.org

252, 2772, 16632, 72072, 252252, 756756, 2018016, 4900896, 11027016, 23279256, 46558512, 88884432, 162954792, 288304632, 494236512, 823727520, 1338557220, 2125943820, 3307023720, 5047562520, 7571343780, 11176745580, 16257084480, 23325382080, 33044291280
Offset: 0

Views

Author

Zerinvary Lajos, Apr 22 2005

Keywords

Examples

			If n=0 then C(0+5,0+0)*C(0+10,5) = C(5,0)*C(10,5) = 1*252 = 252.
If n=4 then C(4+5,4+0)*C(4+10,5) = C(9,4)*C(14,5) = 126*2002 = 252252.

Links

Crossrefs

Cf. A062190.

Programs

  • Magma
    [Binomial(n+5,n)*Binomial(n+10,5): n in [0..30]]; // G. C. Greubel, Nov 25 2017
  • Mathematica
    Table[Binomial[n+5,n]Binomial[n+10,5],{n,0,20}] (* Harvey P. Dale, Feb 06 2015 *)
  • PARI
    Vec(252/(1-x)^11 + O(x^100)) \\ Colin Barker, Feb 07 2015

Formula

G.f.: 252 / (1-x)^11. - Colin Barker, Feb 07 2015
a(n) = A000389(n+5)*A000389(n+10) = 252*A001287(n+11). - R. J. Mathar, Nov 29 2015
From Amiram Eldar, Aug 30 2022: (Start)
Sum_{n>=0} 1/a(n) = 5/1134.
Sum_{n>=0} (-1)^n/a(n) = 1280*log(2)/63 - 447047/31752. (End)

Extensions

Corrected and extended by Harvey P. Dale, Feb 06 2015

A113894 a(n) = binomial(2*n, n) * binomial(2*n+5, n).

Original entry on oeis.org

1, 14, 216, 3300, 50050, 756756, 11435424, 172931616, 2618916300, 39731777800, 603923022560, 9197348345640, 140334704005860, 2145155731845000, 32847841152720000, 503812888080163200, 7739375660195721300, 119063197872768778200, 1834199559285273180000
Offset: 0

Views

Author

Zerinvary Lajos, Jan 28 2006

Keywords

Examples

			a(0) = C(0,0)*C(5,0) = 1*1 = 1.
a(4) = C(8,4)*C(13,4) = 70*715 = 50050.
a(10) = C(20,10)*C(25,10) = 184756*3268760 = 603923022560.

Links

Crossrefs

Cf. A062190.

Programs

  • Magma
    A113894:= func< n | (n+1)*Catalan(n)*Binomial(2*n+5,n) >;
[A113894(n): n in [0..30]]; // G. C. Greubel, Mar 01 2025
  • Mathematica
    Table[Binomial[2n,n]Binomial[2n+5,n],{n,0,20}] (* Harvey P. Dale, Apr 11 2020 *)
  • PARI
    a(n) = {binomial(2*n, n) * binomial(5+2*n, n)} \\ Andrew Howroyd, Jan 07 2020
  • SageMath
    def A113894(n): return binomial(2*n,n)*binomial(2*n+5,n)
print([A113894(n) for n in range(31)]) # G. C. Greubel, Mar 01 2025

Formula

a(n) = A062190(2*n, n).

Extensions

Name edited and terms a(13) and beyond from Andrew Howroyd, Jan 07 2020

A027823 a(n) = 77*(n+1)*binomial(n+6,11).

Original entry on oeis.org

462, 6468, 48048, 252252, 1051050, 3699696, 11435424, 31855824, 81477396, 193993800, 434546112, 923410488, 1873980108, 3651858672, 6864396000, 12493200720, 22086194130, 38030772780, 63935791920, 105157552500, 169513974630, 268241893920, 417265168320
Offset: 5

Views

Author

Thi Ngoc Dinh (via R. K. Guy)

Keywords

Comments

Number of 18-subsequences of [ 1, n ] with just 6 contiguous pairs.

Links

Crossrefs

Cf. A062190.

Programs

  • Mathematica
    Table[77(n+1) Binomial[n+6,11],{n,5,40}] (* or *) LinearRecurrence[{13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1},{462,6468,48048,252252,1051050,3699696,11435424,31855824,81477396,193993800,434546112,923410488,1873980108},30] (* Harvey P. Dale, Oct 20 2016 *)

Formula

G.f.: 462*(1+x)*x^5/(1-x)^13.
a(n) = C(n+1, 6)*C(n+6, 6). - Zerinvary Lajos, Jun 08 2005; corrected by R. J. Mathar, Feb 13 2016
From Amiram Eldar, Feb 04 2022: (Start)
Sum_{n>=5} 1/a(n) = 10446403/176400 - 6*Pi^2.
Sum_{n>=5} (-1)^(n+1)/a(n) = 3*Pi^2 - 82899/2800. (End)

A104680 a(n) = binomial(n+7,7)*binomial(n+12,7).

Original entry on oeis.org

792, 13728, 123552, 772200, 3775200, 15402816, 54609984, 172931616, 498841200, 1330243200, 3316739712, 7801876368, 17439488352, 37263864000, 76488984000, 151448188320, 290275694280, 540192201120, 978609060000, 1729734435000, 2988981103680, 5058275713920
Offset: 0

Views

Author

Zerinvary Lajos, Apr 22 2005

Keywords

Examples

			If n=0 then C(0+7,0+0)*C(0+12,7) = C(7,0)*C(12,7) = 1*792 = 792.
If n=6 then C(6+7,6+0)*C(6+12,7) = C(13,6)*C(18,7) = 1716*32824 = 54609984.

Links

Crossrefs

Cf. A062190.

Programs

  • Mathematica
    a[n_] := Binomial[n + 7, 7] * Binomial[n + 12, 7]; Array[a, 25, 0] (* Amiram Eldar, Aug 30 2022 *)

Formula

G.f.: -264*(3+7*x+3*x^2)/(x-1)^15. - R. J. Mathar, Nov 29 2015
From Amiram Eldar, Aug 30 2022: (Start)
Sum_{n>=0} 1/a(n) = 1263966463/1306800 - 98*Pi^2.
Sum_{n>=0} (-1)^n/a(n) = 3935051/13068 - 14336*log(2)/33. (End)
Previous Showing 11-20 of 28 results. Next

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