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 19 results. Next

A070782 a(n) = Sum_{k=0..n} binomial(5*n,5*k).

Original entry on oeis.org

1, 2, 254, 6008, 215766, 6643782, 215492564, 6863694378, 219993856006, 7035859329512, 225191238869774, 7205634556190798, 230585685502492596, 7378682274243863442, 236118494435702913134, 7555784484021765207768, 241785184867484394069286, 7737125013254912900576822
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), this sequence (b=5), A070967 (b=6), A094211 (b=7), A070832 (b=8), A094213 (b=9), A070833 (b=10).

Programs

  • Mathematica
    LinearRecurrence[{21,353,-32},{1,2,254},20] (* Harvey P. Dale, Jun 18 2023 *)
  • PARI
    a(n)=sum(k=0,n,binomial(5*n,5*k))
  • PARI
    Vec((1 - 19*x - 141*x^2) / ((1 - 32*x)*(1 + 11*x - x^2)) + O(x^20)) \\ Colin Barker, May 27 2019

Formula

a(n) = (1/5)*32^n + (2/5)*(-11/2 + (5/2)*sqrt(5))^n + (2/5)*(-11/2 - (5/2)*sqrt(5))^n.
Let b(n) = a(n) - 2^(5n)/5; then b(n) + 11*b(n-1) - b(n-2) = 0. - Benoit Cloitre, May 27 2004
From Colin Barker, May 27 2019: (Start)
G.f.: (1 - 19*x - 141*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.
(End)

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

A090407 a(n) = Sum_{k = 0..n} C(4*n + 1, 4*k).

Original entry on oeis.org

1, 6, 136, 2016, 32896, 523776, 8390656, 134209536, 2147516416, 34359607296, 549756338176, 8796090925056, 140737496743936, 2251799780130816, 36028797153181696, 576460751766552576, 9223372039002259456
Offset: 0

Views

Author

Paul Barry, Nov 29 2003

Keywords

Links

Crossrefs

Cf. A070775, A001025, A090408, A038503.

Programs

  • Mathematica
    Table[Sum[Binomial[4n+1,4k],{k,0,n}],{n,0,30}] (* or *) LinearRecurrence[ {12,64},{1,6},30] (* Harvey P. Dale, Jan 19 2012 *)

Formula

From Harvey P. Dale, Jan 19 2012: (Start)
a(0) = 1, a(1) = 6, a(n) = 12*a(n-1)+64*a(n-2).
G.f.: (6*x-1)/(64*x^2+12*x-1). (End)
a(n) = (1/2) * 4^n * (4^n + (-1)^n). - Peter Bala, Feb 06 2019

A090408 a(n) = Sum_{k=0..n} binomial(4n+3,4k).

Original entry on oeis.org

1, 36, 496, 8256, 130816, 2098176, 33550336, 536887296, 8589869056, 137439215616, 2199022206976, 35184376283136, 562949936644096, 9007199321849856, 144115187807420416, 2305843010287435776, 36893488143124135936, 590295810375885520896, 9444732965670570950656
Offset: 0

Views

Author

Paul Barry, Nov 29 2003

Keywords

Links

Crossrefs

Cf. A070775, A001025, A090407, A038503.

Programs

  • Mathematica
    Table[Sum[Binomial[4n+3,4k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Feb 12 2020 *)

Formula

a(n) = 2^(1+4*n) - (-1)^n*4^n. - R. J. Mathar, Nov 27 2014

A177808 Triangle T(n,m) = binomial(4*n, 4*m), 0 <= m <= n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 70, 1, 1, 495, 495, 1, 1, 1820, 12870, 1820, 1, 1, 4845, 125970, 125970, 4845, 1, 1, 10626, 735471, 2704156, 735471, 10626, 1, 1, 20475, 3108105, 30421755, 30421755, 3108105, 20475, 1, 1, 35960, 10518300, 225792840, 601080390, 225792840, 10518300, 35960, 1, 1, 58905, 30260340, 1251677700, 7307872110, 7307872110, 1251677700, 30260340, 58905, 1
Offset: 0

Views

Author

Roger L. Bagula, Dec 13 2010

Keywords

Comments

Row sums are A070775(n).

Examples

			1;
1, 1;
1, 70, 1;
1, 495, 495, 1;
1, 1820, 12870, 1820, 1;
1, 4845, 125970, 125970, 4845, 1;
1, 10626, 735471, 2704156, 735471, 10626, 1;
1, 20475, 3108105, 30421755, 30421755, 3108105, 20475, 1;
1, 35960, 10518300, 225792840, 601080390, 225792840, 10518300, 35960, 1;
1, 58905, 30260340, 1251677700, 7307872110, 7307872110, 1251677700,30260340, 58905, 1;
1, 91390, 76904685, 5586853480, 62852101650, 137846528820, 62852101650, 5586853480, 76904685, 91390, 1;

Crossrefs

Cf. A086645, A034839, A070775.

Programs

  • Maple
    A177808 := proc(n,m) binomial(4*n,4*m) ; end proc: # R. J. Mathar, Dec 13 2010
  • Mathematica
    t[n_, m_] = Binomial[n, 4*m];
Table[Table[t[n, m], {m, 0, Floor[n/4]}], {n, 0, 40, 4}];
Flatten[%]

Formula

Right-left symmetric: T(n,m) = T(n,n-m).

A139459 Triangle read by rows: binomial(3*n,3*k), 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 20, 1, 1, 84, 84, 1, 1, 220, 924, 220, 1, 1, 455, 5005, 5005, 455, 1, 1, 816, 18564, 48620, 18564, 816, 1, 1, 1330, 54264, 293930, 293930, 54264, 1330, 1, 1, 2024, 134596, 1307504, 2704156, 1307504, 134596, 2024, 1, 1, 2925, 296010, 4686825, 17383860, 17383860, 4686825, 296010, 2925, 1
Offset: 0

Views

Author

Gary W. Adamson, Apr 22 2008

Keywords

Comments

ConvOffsStoT transform of the dodecahedral numbers A006566 starting (1, 20, 84, 220,...).
Row sums give A007613.
The matrix inverse starts:
1;
-1,1;
19,-20,1;
-1513,1596,-84,1;
315523,-332860,17556,-220,1;
-136085041,143562965,-7572565,95095,-455,1;
105261234643,-111045393456,5857368972,-73562060,352716,-816,1; - R. J. Mathar, Mar 22 2013

Examples

			First few rows of the triangle are:
  [0] 1;
  [1] 1,   1;
  [2] 1,  20,     1;
  [3] 1,  84,    84,     1;
  [4] 1, 220,   924,   220,     1;
  [5] 1, 455,  5005,  5005,   455,   1;
  [6] 1, 816, 18564, 48620, 18564, 816, 1;
  ...
Row 5 = (1, 220, 924, 220, 1) = ConvOffs transform of (1, 20, 84, 220); where A006566 = (0, 1, 20, 84, 220, 455, ...).

Crossrefs

Cf. A006566, A007613, A086645, A034839, A070775, A177808.

Programs

  • Mathematica
    Table[Binomial[3*n, 3*k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 01 2025 *)

Extensions

More terms from Amiram Eldar, Jun 01 2025
Showing 1-10 of 19 results. Next

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