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 1-3 of 3 results.

A159841 Triangle T(n,k) = binomial(3*n+1, 2*n+k+1), read by rows.

Original entry on oeis.org

1, 4, 1, 21, 7, 1, 120, 45, 10, 1, 715, 286, 78, 13, 1, 4368, 1820, 560, 120, 16, 1, 27132, 11628, 3876, 969, 171, 19, 1, 170544, 74613, 26334, 7315, 1540, 231, 22, 1, 1081575, 480700, 177100, 53130, 12650, 2300, 300, 25, 1, 6906900, 3108105, 1184040, 376740
Offset: 0

Views

Author

Philippe Deléham, Apr 23 2009

Keywords

Comments

T(n,0) = A045721(n), T(2n,n) = A079590(n).

Examples

			Triangle begins:
     1;
     4,    1;
    21,    7,    1;
   120,   45,   10,    1;
   715,  286,   78,   13,    1;
  4368, 1820,  560,  120,   16,    1;
  ...

Links

Crossrefs

Cf. A045721, A079590.

Programs

  • Magma
    /* As triangle */ [[Binomial(3*n+1, 2*n+k+1): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 19 2018
  • Mathematica
    f[n_,k_]:=Binomial[3n+1,2n+k+1]; Table[ f[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Robert G. Wilson v, May 31 2009 *)
  • PARI
    for(n=0,10, for(k=0,n, print1(binomial(3*n+1, 2*n+k+1), ", "))) \\ G. C. Greubel, May 19 2018

Formula

T(n,0) = 4*T(n-1,0) + 5*T(n-1,1) + T(n-1,2), T(n+1,k+1) = T(n,k) + 3*T(n,k+1) + 3*T(n,k+2) + T(n,k+3) for k >= 0.

Extensions

More terms from Robert G. Wilson v, May 31 2009

A385497 a(n) = Sum_{k=0..n} binomial(6*n+1,k).

Original entry on oeis.org

1, 8, 92, 1160, 15276, 206368, 2835200, 39419864, 553000876, 7811733392, 110962066532, 1583318009160, 22677731944032, 325849065291056, 4694837606889424, 67803714186207280, 981265566082447276, 14227018304102548368, 206608052310739404392, 3004777578508008253808
Offset: 0

Views

Author

Seiichi Manyama, Aug 17 2025

Keywords

Links

Crossrefs

Cf. A000302, A160906, A386811, A386812.
Cf. A004355, A079590, A079679, A226705.
Cf. A002295, A385719.

Programs

  • Magma
    [&+[Binomial(6*n+1, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 18 2025
  • Mathematica
    Table[Sum[Binomial[6*n+1,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 18 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(6*n+1, k));

Formula

a(n) = [x^n] (1+x)^(6*n+1)/(1-x).
a(n) = [x^n] 1/((1-x)^(5*n+1) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(6*n+1,k) * binomial(6*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(6*n-k,n-k).
G.f.: 1/(1 - 4*x*g^4*(3-g)) where g = 1+x*g^6 is the g.f. of A002295.
G.f.: g^2/((2-g) * (6-5*g)) where g = 1+x*g^6 is the g.f. of A002295.
G.f.: B(x)^2/(1 + 2*(B(x)-1)/3), where B(x) is the g.f. of A004355.
a(n) ~ 2^(6*n-1) * 3^(6*n + 3/2) / (sqrt(Pi*n) * 5^(5*n + 1/2)). - Vaclav Kotesovec, Aug 19 2025
D-finite with recurrence +5*n*(5*n-3) *(25275337086729240289198339046875*n +471647298106881091699147254457046) *(5*n-1)*(5*n-4)*(5*n-2)*a(n) +(78985428396028875903744809521484375*n^6 -559942234844855804767211877804090453801*n^5 +3587636672285250929619857349305543417315*n^4 -10153151347942687598200945831585305558855*n^3 +14794114656715293872778407292185015920550*n^2 -10846691360081598422810600143797325763664*n +3179147242764665659301361496311050364480)*a(n-1) +40*(916451705547792050816664342989042382392*n^6 -15754440652132350078674083937326518806004*n^5 +117614110896134855700514819789186651267682*n^4 -471111363407608954402735569277858473721059*n^3 +1053743992048348087929158710510276422876431*n^2 -1242809524683997363700671579060256757555078*n +603414490131980309336751304501155726403152) *a(n-2) +3072*(-950768355029313182341332806167821761828*n^6 +17097100921628721474237101055297828968024*n^5 -128090998271831890487248970509140383514230*n^4 +509544263618626898681417576914870842148685*n^3 -1132270964907780344616429736070172799129247*n^2 +1330655887974191637410201798934319046990726*n -645481184978535641217111809931780144149880) *a(n-3) +884736*(3*n-11) *(6*n-17) *(61801507754400081418308631750717123*n -123657551673181017806623428016627104) *(6*n-19)*(3*n-10)*(2*n-7)*a(n-4)=0. - R. J. Mathar, Aug 26 2025

A387091 a(n) = binomial(9*n+1,n).

Original entry on oeis.org

1, 10, 171, 3276, 66045, 1370754, 28989675, 621216192, 13442126049, 293052087900, 6426898010533, 141629804643600, 3133614810784185, 69566517009302868, 1548833316392624625, 34569147570568156800, 773240476721553042345, 17328840976366636057110
Offset: 0

Views

Author

Seiichi Manyama, Aug 16 2025

Keywords

Links

Crossrefs

Cf. A001700, A045721, A052203, A079589, A079590.
Cf. A062994, A169958.

Programs

  • Mathematica
    A387091[n_] := Binomial[9*n + 1, n]; Array[A387091, 20, 0] (* Paolo Xausa, Aug 20 2025 *)
  • PARI
    a(n) = binomial(9*n+1, n);

Formula

a(n) = Sum_{k=0..n} binomial(9*n-k,n-k).
G.f.: 1/(1 - x*g^7*(9+g)) where g = 1+x*g^9 is the g.f. of A062994.
G.f.: g^2/(9-8*g) where g = 1+x*g^9 is the g.f. of A062994.
G.f.: B(x)^2/(1 + 8*(B(x)-1)/9), where B(x) is the g.f. of A169958.
D-finite with recurrence +128*n*(8*n-5)*(4*n-1)*(8*n+1)*(2*n-1)*(8*n-1)*(4*n-3)*(8*n-3)*a(n) -81*(9*n-7)*(9*n-5)*(3*n-1)*(9*n-1)*(9*n+1)*(3*n-2)*(9*n-4)*(9*n-2)*a(n-1)=0. - R. J. Mathar, Aug 19 2025
a(n) ~ 3^(18*n+3) / (sqrt(Pi*n) * 2^(24*n+5)). - Vaclav Kotesovec, Aug 20 2025
Showing 1-3 of 3 results.

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