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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A132688 a(n) = binomial(2^n + 3*n, n).

Original entry on oeis.org

1, 5, 45, 680, 20475, 1533939, 350161812, 280384608504, 847073824772175, 9894081531608130857, 446730013630787463700695, 77328499046923986969058944720, 50891283683781760304442885961988100
Offset: 0

Views

Author

Paul D. Hanna, Aug 26 2007

Keywords

Links

Crossrefs

Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), A060690 (1,-1), A132683 (1,0), A132684 (1,1), A132685 (2,0), A132686 (2,1), A132687 (3,-1), this sequence (3,0), A132689 (3,1).
Cf. A136555.

Programs

  • Magma
    [Binomial(2^n +3*n, n): n in [0..20]]; // G. C. Greubel, Mar 13 2021
  • Mathematica
    Table[Binomial[2^n+3n,n],{n,0,20}] (* Harvey P. Dale, Oct 30 2018 *)
  • PARI
    a(n)=binomial(2^n+3*n,n)
  • Sage
    [binomial(2^n +3*n, n) for n in (0..20)] # G. C. Greubel, Mar 13 2021

Formula

a(n) = [x^n] 1/(1-x)^(2^n + 2*n + 1).

A132689 a(n) = binomial(2^n + 3*n + 1, n).

Original entry on oeis.org

1, 6, 55, 816, 23751, 1712304, 377447148, 294109729200, 871896500955975, 10061777828754031380, 451004941990890693018405, 77739225019650285306412710240, 51039474754930845750609669420261300
Offset: 0

Views

Author

Paul D. Hanna, Aug 26 2007

Keywords

Links

Crossrefs

Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), A060690 (1,-1), A132683 (1,0), A132684 (1,1), A132685 (2,0), A132686 (2,1), A132687 (3,-1), A132688 (3,0), this sequence (3,1).
Cf. A136555.

Programs

  • Magma
    [Binomial(2^n+3*n+1, n) : n in [0..15]]; // Wesley Ivan Hurt, Nov 20 2014
  • Maple
    A132689:=n->binomial(2^n+3*n+1, n): seq(A132689(n), n=0..15); # Wesley Ivan Hurt, Nov 20 2014
  • Mathematica
    Table[Binomial[2^n +3n +1, n], {n, 0, 15}] (* Wesley Ivan Hurt, Nov 20 2014 *)
  • PARI
    a(n)=binomial(2^n+3*n+1,n)
  • Sage
    [binomial(2^n +3*n+1, n) for n in (0..15)] # G. C. Greubel, Feb 15 2021

Formula

a(n) = [x^n] 1/(1-x)^(2^n + 2*n + 2).

A136507 a(n) = Sum_{k=0..n} binomial(2^(n-k) + k, n-k).

Original entry on oeis.org

1, 3, 10, 71, 1925, 203904, 75214965, 94608676477, 409763735870986, 6208539881584781823, 334272186911271376874561, 64832512634295914941490910360, 45811927207957062190019240099653265
Offset: 0

Views

Author

Paul D. Hanna, Jan 01 2008

Keywords

Links

Crossrefs

Cf. A014070 (C(2^n, n)), A136505 (C(2^n+1, n)), A136506 (C(2^n+2, n)).
Cf. A136508, A136509, A136555.

Programs

  • Magma
    [(&+[Binomial(2^k +n-k, k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Mar 14 2021
  • Maple
    A136507:= n-> add(binomial(2^k +n-k, k), k=0..n); seq(A136507(n), n=0..20); # G. C. Greubel, Mar 14 2021
  • Mathematica
    Table[Sum[Binomial[2^(n-k)+k,n-k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Mar 08 2015 *)
  • PARI
    {a(n)=sum(k=0,n,binomial(2^(n-k)+k,n-k))}
for(n=0,16, print1(a(n),", "))
  • PARI
    /* a(n) = coefficient of x^n in o.g.f. series: */
{a(n)=polcoeff(sum(i=0,n,1/(1-x-2^i*x^2 +x*O(x^n))*log(1+2^i*x +x*O(x^n))^i/i!),n)}
for(n=0,16, print1(a(n),", "))
  • Sage
    [sum(binomial(2^k +n-k, k) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Mar 14 2021

Formula

G.f.: A(x) = Sum_{n>=0} (1 - x - 2^n*x^2)^(-1) * log(1 + 2^n*x)^n/n!.
a(n) ~ 2^(n^2) / n!. - Vaclav Kotesovec, Jul 02 2016
a(n) = Sum_{k=0..n} A136555(n-k+1, k). - G. C. Greubel, Mar 14 2021

A132625 Triangle T, read by rows, where row n+1 of T = row n of T^(2^n) with appended '1' for n>=0 with T(0,0)=1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 14, 4, 1, 1, 336, 60, 8, 1, 1, 25836, 2960, 248, 16, 1, 1, 6251504, 454072, 24800, 1008, 32, 1, 1, 4838830976, 216266368, 7603952, 202944, 4064, 64, 1, 1, 12344615283200, 328381917376, 7190266752, 124427232, 1641856, 16320, 128, 1, 1
Offset: 0

Views

Author

Paul D. Hanna, Aug 25 2007, Jan 07 2008

Keywords

Comments

Let R_{n} equal row n of square array A136555, where A136555(n,k) = C(2^k + n-1, k); this triangle transforms rows of A136555: T * R_{n} = R_{n+1} for n>=0.

Examples

			Triangle begins:
1;
1, 1;
2, 1, 1;
14, 4, 1, 1;
336, 60, 8, 1, 1;
25836, 2960, 248, 16, 1, 1;
6251504, 454072, 24800, 1008, 32, 1, 1;
4838830976, 216266368, 7603952, 202944, 4064, 64, 1, 1;
12344615283200, 328381917376, 7190266752, 124427232, 1641856, 16320, 128, 1, 1; ...
GENERATE T FROM MATRIX POWERS OF T.
Matrix power T^4 begins:
1;
4, 1;
14, 4, 1; <-- row 3 of T
96, 22, 4, 1;
1941, 316, 38, 4, 1;
129206, 14185, 1140, 70, 4, 1; ...
where row 3 of T = row 2 of T^(2^2) with appended '1'.
Matrix power T^8 begins:
1;
8, 1;
44, 8, 1;
336, 60, 8, 1; <-- row 4 of T
6062, 872, 92, 8, 1;
345596, 35734, 2712, 156, 8, 1; ...
where row 4 of T = row 3 of T^(2^3) with appended '1'.
Matrix power T^16 begins:
1;
16, 1;
152, 16, 1;
1504, 184, 16, 1;
25836, 2960, 248, 16, 1; <-- row 5 of T
1197304, 109500, 7408, 376, 16, 1; ...
where row 5 of T = row 4 of T^(2^4) with appended '1'.
Alternate generating method:
RoW 3: start with '1' followed by (2^2 - 1) zeros;
take partial sums and append (2^1 - 1) zero;
take partial sums twice more:
(1), 0, 0, 0;
1, 1, 1, (1), 0;
1, 2, 3, 4, (4);
1, 3, 6, 10, (14);
the final nonzero terms form row 3: [14, 4, 1, 1].
Row 4: start with '1' followed by (2^3 - 1) zeros;
take partial sums and append (2^2 - 1) zeros;
take partial sums and append (2^1 - 1) zero;
take partial sums twice more:
(1), 0, 0, 0, 0, 0, 0, 0;
1, 1, 1, 1, 1, 1, 1, (1), 0, 0, 0;
1, 2, 3, 4, 5, 6, 7, 8, 8, 8, (8), 0;
1, 3, 6, 10, 15, 21, 28, 36, 44, 52, 60, (60);
1, 4, 10, 20, 35, 56, 84, 120, 164, 216, 276, (336);
the final nonzero terms form row 4: [336, 60, 8, 1, 1].
Continuing in this way produces all the rows of this triangle.

Crossrefs

Cf. variants: A101479, A132610, A132615; columns: A132626, A132627.
Cf. A136555.

Programs

  • PARI
    T(n, k)=local(A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i, j]=1, B[i, j]=(A^(2^(i-2)))[i-1, j]); )); A=B); return( ((A)[n+1, k+1]))
  • PARI
    /* Generate using partial sums method (faster) */ T(n, k)=local(A=vector(n+1), p); A[1]=1; for(j=1, n-k, p=2^n-2^(n-j)-j; A=Vec((Polrev(A)+x*O(x^p))/(1-x))); A[p+1]
  • PARI
    /* As Row Transformation of Square Array A136555(n,k) = C(2^k + n-1, k): */ T(n,k)=local(M=matrix(n+2,n+2,r,c,binomial(2^(c-1)+r-2,c-1)), N=matrix(n+1,n+1,r,c,M[r,c]),P=matrix(n+1,n+1,r,c,M[r+1,c]),R=P~*N~^-1); R[n+1,k+1]

A136557 a(n) = Sum_{k=0..n} binomial(2^k + n-k-1, k).

Original entry on oeis.org

1, 2, 6, 45, 1436, 171836, 68149425, 89431630740, 396956313475102, 6099399658235428041, 331007760926212498510464, 64484289650612910347505873728, 45677712418497545460138258802186905
Offset: 0

Views

Author

Paul D. Hanna, Jan 07 2008

Keywords

Links

Crossrefs

Cf. A136555, A136556.

Programs

  • Magma
    [(&+[Binomial(2^k +n-k-1, k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Mar 15 2021
  • Maple
    A136557:= n-> add(binomial(2^k +n-k-1, k), k=0..n); seq(A136557(n), n=0..20); # G. C. Greubel, Mar 15 2021
  • Mathematica
    Table[Sum[Binomial[2^k+n-k-1, k], {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Jul 02 2016 *)
  • PARI
    a(n)=sum(k=0,n,binomial(2^k+n-k-1,k))
  • PARI
    /* As coefficient of x^n in the g.f.: */ {a(n)=polcoeff(sum(i=0,n,((1+2^i*x+x*O(x^n))*(1-x-2^i*x^2))^-1*log(1+2^i*x+x*O(x^n))^i/i!),n)}
  • Sage
    [sum(binomial(2^k +n-k-1, k) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Mar 15 2021

Formula

Equals antidiagonal sums of square array A136555.
G.f.: A(x) = Sum_{n>=0} (1+2^n*x)^-1 * (1-x-2^n*x^2)^-1 * log(1+2^n*x)^n / n!.
a(n) ~ 2^(n^2) / n!. - Vaclav Kotesovec, Jul 02 2016
Previous Showing 11-15 of 15 results.

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