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.

Showing 1-10 of 15 results. Next

A070775 a(n) = Sum_{k=0..n} binomial(4*n,4*k).

Original entry on oeis.org

1, 2, 72, 992, 16512, 261632, 4196352, 67100672, 1073774592, 17179738112, 274878431232, 4398044413952, 70368752566272, 1125899873288192, 18014398643699712, 288230375614840832, 4611686020574871552, 73786976286248271872, 1180591620751771041792, 18889465931341141901312
Offset: 0

Views

Author

Sebastian Gutierrez and Sarah Kolitz (skolitz(AT)mit.edu), May 15 2002

Keywords

Comments

Also the cogrowth sequence of the 16-element group C4 X C4 = . - Sean A. Irvine, Nov 09 2024

Links

Crossrefs

Sum_{k=0..n} binomial(b*n,b*k): A000079 (b=1), A081294 (b=2), A007613 (b=3), this sequence (b=4), A070782 (b=5), A070967 (b=6), A094211 (b=7), A070832 (b=8), A094213 (b=9), A070833 (b=10).

Programs

  • Maple
    a := n -> if n = 0 then 1 else 4^(n - 1)*(2*(-1)^n + 4^n) fi:
seq(a(n), n = 0..19); # Peter Luschny, Jul 02 2022
  • Mathematica
    Table[Sum[Binomial[4n,4k],{k,0,n}],{n,0,30}] (* or *) Join[{1}, LinearRecurrence[{12,64},{2,72},30]] (* Harvey P. Dale, Apr 24 2011 *)
  • PARI
    a(n)=sum(k=0,n,binomial(4*n,4*k))
  • PARI
    N=66; x='x+O('x^N); Vec((1-10*x-16*x^2)/((1-16*x)*(1+4*x))) \\ Seiichi Manyama, Mar 15 2019

Formula

a(n) = (1/2)*(-4)^n + (1/4)*16^n for n > 0.
Let b(n) = a(n) - 2^(4n)/4 then b(n+1) = 4*b(n) - Benoit Cloitre, May 27 2004
G.f.: (1 - 10*x - 16*x^2)/((1-16*x)*(1+4*x)). - Seiichi Manyama, Mar 15 2019
G.f.: ((cos(x) + cosh(x))/2)^2 = Sum_{n >= 0} a(n)*x(4*n)/(4*n)!. - Peter Bala, Jun 20 2022

A070832 a(n) = Sum_{k=0..n} binomial(8*n,8*k).

Original entry on oeis.org

1, 2, 12872, 1470944, 622116992, 125858012672, 36758056208384, 8793364151263232, 2334899414608412672, 586347560750962049024, 151652224498623981289472, 38612725801339748322639872, 9913426188311626771400228864
Offset: 0

Views

Author

Sebastian Gutierrez and Sarah Kolitz (skolitz(AT)mit.edu), May 15 2002

Keywords

Links

Crossrefs

Sum_{k=0..n} binomial(b*n,b*k): A000079 (b=1), A081294 (b=2), A007613 (b=3), A070775 (b=4), A070782 (b=5), A070967 (b=6), A094211 (b=7), this sequence (b=8), A094213 (b=9), A070833 (b=10).

Programs

  • Mathematica
    Table[Sum[Binomial[8n,8k],{k,0,n}],{n,0,15}] (* Harvey P. Dale, Nov 25 2020 *)
  • PARI
    a(n)=sum(k=0,n,binomial(8*n,8*k)); \\ Benoit Cloitre, May 27 2004
  • PARI
    Vec((1 - 134*x - 20280*x^2 + 207296*x^3 + 8192*x^4) / ((1 - 16*x)*(1 - 256*x)*(1 + 136*x + 16*x^2)) + O(x^15)) \\ Colin Barker, May 27 2019

Formula

Let b(n) = a(n)-2^(8*n)/8 then b(n)+120*b(n-1)-2160*b(n-2)-256*b(n-3)=0. - Benoit Cloitre, May 27 2004
a(n) = 1/4*16^n + 1/8*256^n + 1/4*(-68 + 48*sqrt(2))^n + 1/4*(-68-48*sqrt(2))^n.
From Colin Barker, May 27 2019: (Start)
G.f.: (1 - 134*x - 20280*x^2 + 207296*x^3 + 8192*x^4) / ((1 - 16*x)*(1 - 256*x)*(1 + 136*x + 16*x^2)).
a(n) = 21*a(n-1) + 353*a(n-2) - 32*a(n-3) for n>4.
(End)

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 15 2007

A070967 a(n) = Sum_{k=0..n} binomial(6*n,6*k).

Original entry on oeis.org

1, 2, 926, 37130, 2973350, 174174002, 11582386286, 729520967450, 47006639297270, 2999857885752002, 192222214478506046, 12295976362284182570, 787111112023373201990, 50370558298891875954002, 3223838658635388303336206, 206322355109994528871954490
Offset: 0

Views

Author

Sebastian Gutierrez and Sarah Kolitz (skolitz(AT)mit.edu), May 16 2002

Keywords

References

  • Matthijs Coster, Supercongruences, Thesis, Jun 08, 1988.

Links

Crossrefs

Sum_{k=0..n} binomial(b*n,b*k): A000079 (b=1), A081294 (b=2), A007613 (b=3), A070775 (b=4), A070782 (b=5), this sequence (b=6), A094211 (b=7), A070832 (b=8), A094213 (b=9), A070833 (b=10).

Programs

  • Mathematica
    Table[Sum[Binomial[6n,6k],{k,0,n}],{n,0,20}] (* or *) LinearRecurrence[ {38,1691,-1728},{1,2,926,37130},30] (* Harvey P. Dale, Jun 19 2021 *)
  • PARI
    a(n)=sum(k=0,n,binomial(6*n,6*k))
  • PARI
    a(n)=if(n<0,0,(2*(-27)^n+2+64^n+0^n)/6)
  • PARI
    a(n)=if(n<0,0,polsym(x*(x-64)*(x+27)^2*(x-1)^2,n)[n+1]/6)

Formula

G.f.: (1-36x-841x^2+288x^3)/((1-x)*(1+27x)*(1-64x)).
a(n) = ((-27)^n + 1)/3 + (64^n + 0^n)/6.
Let b(n) = a(n)-2^(6n)/6 then b(n)+26*b(n-1)-27*b(n-2) = 0. - Benoit Cloitre, May 27 2004

A070833 a(n) = Sum_{k=0..n} binomial(10*n,10*k).

Original entry on oeis.org

1, 2, 184758, 60090032, 139541849878, 94278969044262, 126648421364527548, 111019250117021378442, 125257104438594491956518, 121088185204450642433930072, 128442558588779813655233443038, 128767440665677943753184267342902
Offset: 0

Views

Author

Sebastian Gutierrez and Sarah Kolitz (skolitz(AT)mit.edu), May 15 2002

Keywords

Links

Crossrefs

Sum_{k=0..n} binomial(b*n,b*k): A000079 (b=1), A081294 (b=2), A007613 (b=3), A070775 (b=4), A070782 (b=5), A070967 (b=6), A094211 (b=7), A070832 (b=8), A094213 (b=9), this sequence (b=10). Cf. A000032.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(10*n, 10*k)); \\ Michel Marcus, Mar 15 2019
  • PARI
    Vec((1 - 520*x - 404083*x^2 + 37605988*x^3 + 117888625*x^4 - 320000*x^5) / ((1 - 1024*x)*(1 - 123*x + x^2)*(1 + 625*x + 3125*x^2)) + O(x^15)) \\ Colin Barker, Mar 15 2019

Formula

a(n) = 1/10*1024^n+1/5*(-625/2+275/2*sqrt(5))^n+1/5*(-625/2-275/2*sqrt(5))^n+1/5*(123/2+55/2*sqrt(5))^n+1/5*(123/2-55/2*sqrt(5))^n.
G.f.: (1 - 520*x - 404083*x^2 + 37605988*x^3 + 117888625*x^4 - 320000*x^5) / ((1 - 1024*x)*(1 - 123*x + x^2)*(1 + 625*x + 3125*x^2)). - Colin Barker, Mar 15 2019
a(2*n) = (2^(20*n-1) + Lucas(20*n) + 5^(5*n)*Lucas(10*n))/5, for n>0 and for Lucas(n) = A000032(n). - Greg Dresden, Feb 04 2023

A094211 a(n) = Sum_{k=0..n} binomial(7*n,7*k).

Original entry on oeis.org

1, 2, 3434, 232562, 42484682, 4653367842, 644032289258, 79450506979090, 10353832741654602, 1313930226050847938, 168883831255263816554, 21573903987107973878962, 2764126124873404346104778, 353643666623193292098680930, 45276535087893983968685884906
Offset: 0

Views

Author

Benoit Cloitre, May 27 2004

Keywords

Links

Crossrefs

Sum_{k=0..n} binomial(b*n,b*k): A000079 (b=1), A081294 (b=2), A007613 (b=3), A070775 (b=4), A070782 (b=5), A070967 (b=6), this sequence (b=7), A070832 (b=8), A094213 (b=9), A070833 (b=10).

Programs

  • Maple
    A094211:=n->add(binomial(7*n,7*k), k=0..n): seq(A094211(n), n=0..20); # Wesley Ivan Hurt, Feb 16 2017
  • Mathematica
    Table[Sum[Binomial[7n,7k],{k,0,n}],{n,0,20}] (* or *) LinearRecurrence[ {71,7585,-36991,-128},{1,2,3434,232562},20] (* Harvey P. Dale, May 06 2012 *)
  • PARI
    a(n)=sum(k=0,n,binomial(7*n,7*k))

Formula

Let b(n) = a(n) - 2^(7n)/7; then b(n) + 57*b(n-1) - 289*b(n-2) - b(n-3) = 0.
a(n) = 71*a(n-1) + 7585*a(n-2) - 36991*a(n-3) - 128*a(n-4); a(0)=1, a(1)=2, a(2)=3434, a(3)=232562. - Harvey P. Dale, May 06 2012
G.f.: (1 - 69*x - 4293*x^2 + 10569*x^3) / ((1 - 128*x)*(1 + 57*x - 289*x^2 - x^3)). - Colin Barker, May 27 2019
a(n) = (1 + 2*(s*(3 - 4*s^2))^(7*n) + 2*(-1)^n*((1 - 2*s^2)^(7*n) + s^(7*n))) * 2^(7*n)/7, where s = sin(Pi/14). - Vaclav Kotesovec, Apr 17 2023

A094213 a(n) = Sum_{k=0..n} binomial(9*n,9*k).

Original entry on oeis.org

1, 2, 48622, 9373652, 9263421862, 3433541316152, 2140802758677844, 984101481334553024, 536617781178725122150, 265166261617029717011822, 138567978655457801631498052, 70126939586658252408697345838, 36144812798331420987905742371116
Offset: 0

Views

Author

Benoit Cloitre, May 27 2004

Keywords

Links

Crossrefs

Sum_{k=0..n} binomial(b*n,b*k): A000079 (b=1), A081294 (b=2), A007613 (b=3), A070775 (b=4), A070782 (b=5), A070967 (b=6), A094211 (b=7), A070832 (b=8), this sequence (b=9), A070833 (b=10).

Programs

  • Mathematica
    Table[Sum[Binomial[9n,9k],{k,0,n}],{n,0,15}] (* Harvey P. Dale, Jul 14 2019 *)
  • PARI
    a(n)=sum(k=0,n,binomial(9*n,9*k))
  • PARI
    Vec((1 - 263*x - 91731*x^2 + 3035380*x^3 + 1547833*x^4) / ((1 + x)*(1 - 512*x)*(1 + 246*x - 13605*x^2 + x^3)) + O(x^15)) \\ Colin Barker, May 27 2019

Formula

Let b(n) = a(n)-2^(9*n)/9 then b(n)+246*b(n-1)-13605*b(n-2)+b(n-3)+(-1)^n*3078=0.
Conjectures from Colin Barker, May 27 2019: (Start)
G.f.: (1 - 263*x - 91731*x^2 + 3035380*x^3 + 1547833*x^4) / ((1 + x)*(1 - 512*x)*(1 + 246*x - 13605*x^2 + x^3)).
a(n) = 265*a(n-1) + 139823*a(n-2) - 6826204*a(n-3) - 6965249*a(n-4) + 512*a(n-5) for n>4. (End)
a(n) ~ (1/9)*exp(n*9*log(2)) (conjecture). - Bill McEachen, Aug 11 2025

A345455 a(n) = Sum_{k=0..n} binomial(5*n+1,5*k).

Original entry on oeis.org

1, 7, 474, 12393, 427351, 13333932, 430470899, 13733091643, 439924466026, 14072420067757, 450374698997499, 14411355379952868, 461170414282959151, 14757375158697584607, 472236871202375365274, 15111570273013075344193, 483570355262634763462351
Offset: 0

Views

Author

Seiichi Manyama, Jun 20 2021

Keywords

Links

Crossrefs

Sum_{k=0..n} binomial(b*n+c,b*k): A082311 (b=3,c=1), A090407 (b=4,c=1), A070782 (b=5,c=0), this sequence (b=5,c=1), A345456 (b=5,c=2), A345457 (b=5,c=3), A345458 (b=5,c=4).
Cf. A139398.

Programs

  • Mathematica
    a[n_] := Sum[Binomial[5*n + 1, 5*k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Jun 20 2021 *)
LinearRecurrence[{21,353,-32},{1,7,474},20] (* Harvey P. Dale, Jul 20 2021 *)
  • PARI
    a(n) = sum(k=0, n, binomial(5*n+1, 5*k));
  • PARI
    my(N=20, x='x+O('x^N)); Vec((1-14*x-26*x^2)/((1-32*x)*(1+11*x-x^2)))

Formula

G.f.: (1 - 14*x - 26*x^2) / ((1 - 32*x)*(1 + 11*x - x^2)).
a(n) = 21*a(n-1) + 353*a(n-2) - 32*a(n-3) for n>2.
a(n) = A139398(5*n+1).
a(n) = 2^(5*n + 2)/10 + ((-475 + 213*sqrt(5))/phi^(5*n) - ( 65 - 33*sqrt(5))*(-1)^n*phi^(5*n)) / (10*(41*sqrt(5)-90)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jun 20 2021

A345456 a(n) = Sum_{k=0..n} binomial(5*n+2,5*k).

Original entry on oeis.org

1, 22, 859, 25773, 843756, 26789257, 859595529, 27481113638, 879683351911, 28146676447417, 900729032983924, 28822936611339453, 922338323835136341, 29514778095285204502, 944473434343229560419, 30223143962480773595093, 967140672636207153780796
Offset: 0

Views

Author

Seiichi Manyama, Jun 20 2021

Keywords

Links

Crossrefs

Sum_{k=0..n} binomial(b*n+c,b*k): A070782 (b=5,c=0), A345455 (b=5,c=1), this sequence (b=5,c=2), A345457 (b=5,c=3), A345458 (b=5,c=4).
Cf. A139398.

Programs

  • Mathematica
    a[n_] := Sum[Binomial[5*n + 2, 5*k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Jun 20 2021 *)
LinearRecurrence[{21,353,-32},{1,22,859},20] (* Harvey P. Dale, Aug 25 2022 *)
  • PARI
    a(n) = sum(k=0, n, binomial(5*n+2, 5*k));
  • PARI
    my(N=20, x='x+O('x^N)); Vec((1+x+44*x^2)/((1-32*x)*(1+11*x-x^2)))

Formula

G.f.: (1 + x + 44*x^2) / ((1 - 32*x)*(1 + 11*x - x^2)).
a(n) = 21*a(n-1) + 353*a(n-2) - 32*a(n-3) for n>2.
a(n) = A139398(5*n+2).
a(n) = 2^(5*n + 3)/10 + ((-295 + 131*sqrt(5))/phi^(5*n) + (115 - 49*sqrt(5))*(-1)^n*phi^(5*n)) / (10*(41*sqrt(5)-90)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jun 20 2021

A345457 a(n) = Sum_{k=0..n} binomial(5*n+3,5*k).

Original entry on oeis.org

1, 57, 1574, 53143, 1669801, 53774932, 1717012749, 54986385093, 1759098789526, 56296324109907, 1801425114687749, 57646238657975068, 1844672594930734801, 59029601136140621857, 1888946370232447241574, 60446293452901248074943
Offset: 0

Views

Author

Seiichi Manyama, Jun 20 2021

Keywords

Links

Crossrefs

Sum_{k=0..n} binomial(b*n+c,b*k): A090408 (b=4,c=3), A070782 (b=5,c=0), A345455 (b=5,c=1), A345456 (b=5,c=2), this sequence (b=5,c=3), A345458 (b=5,c=4).
Cf. A139398.

Programs

  • Mathematica
    a[n_] := Sum[Binomial[5*n + 3, 5*k], {k, 0, n}]; Array[a, 16, 0] (* Amiram Eldar, Jun 20 2021 *)
  • PARI
    a(n) = sum(k=0, n, binomial(5*n+3, 5*k));
  • PARI
    my(N=20, x='x+O('x^N)); Vec((1+36*x+24*x^2)/((1-32*x)*(1+11*x-x^2)))

Formula

G.f.: (1 + 36*x + 24*x^2) / ((1 - 32*x)*(1 + 11*x - x^2)).
a(n) = 21*a(n-1) + 353*a(n-2) - 32*a(n-3) for n>2.
a(n) = A139398(5*n+3).
a(n) = 2^(5*n + 4)/10 + (( 475 - 213*sqrt(5))/phi^(5*n) + ( 65 - 33*sqrt(5))*(-1)^n*phi^(5*n)) / (10*(41*sqrt(5)-90)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jun 20 2021

A345458 a(n) = Sum_{k=0..n} binomial(5*n+4,5*k).

Original entry on oeis.org

1, 127, 3004, 107883, 3321891, 107746282, 3431847189, 109996928003, 3517929664756, 112595619434887, 3602817278095399, 115292842751246298, 3689341137121931721, 118059247217851456567, 3777892242010882603884, 120892592433742197034643
Offset: 0

Views

Author

Seiichi Manyama, Jun 20 2021

Keywords

Links

Crossrefs

Sum_{k=0..n} binomial(b*n+c,b*k): A070782 (b=5,c=0), A345455 (b=5,c=1), A345456 (b=5,c=2), A345457 (b=5,c=3), this sequence (b=5,c=4).
Cf. A139398.

Programs

  • Mathematica
    a[n_] := Sum[Binomial[5*n + 4, 5*k], {k, 0, n}]; Array[a, 16, 0] (* Amiram Eldar, Jun 20 2021 *)
Total/@Table[Binomial[5n+4,5k],{n,0,20},{k,0,n}] (* or *) LinearRecurrence[{21,353,-32},{1,127,3004},30] (* Harvey P. Dale, Oct 29 2023 *)
  • PARI
    a(n) = sum(k=0, n, binomial(5*n+4, 5*k));
  • PARI
    my(N=20, x='x+O('x^N)); Vec((1+106*x-16*x^2)/((1-32*x)*(1+11*x-x^2)))

Formula

G.f.: (1 + 106*x - 16*x^2) / ((1 - 32*x)*(1 + 11*x - x^2)).
a(n) = 21*a(n-1) + 353*a(n-2) - 32*a(n-3) for n>2.
a(n) = A139398(5*n+4).
a(n) = 2^(5*n + 5)/10 + ((2015 - 901*sqrt(5))/phi^(5*n) - (35 + sqrt(5))*(-1)^n*phi^(5*n)) / (10*(41*sqrt(5)-90)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jun 20 2021
Showing 1-10 of 15 results. Next

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