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 1-4 of 4 results.

A151709 Row sums of A156741.

Original entry on oeis.org

1, 2, 192, 14632, 5451140, 2216555772, 2201283594512, 2563699840815752, 5239330894956743702, 12738172416005805235262, 45354957806572334315266802, 190794310975336315988205573422, 1056059186013450690759502943569093, 6805676661977149073551721890947184830
Offset: 0

Views

Author

N. J. A. Sloane, Jun 06 2009

Keywords

Links

Crossrefs

Cf. A156741.

Programs

  • Mathematica
    A156741[n_, k_]:= Round[Product[Binomial[2*(n+j), 2*(k+j)]/Binomial[2*(n-k+j), 2*j], {j, 0, 8}]];
A151709[n_]:= A151709[n]= Sum[A156741[n, k], {k,0,n}];
Table[A151709[n], {n, 0, 30}] (* G. C. Greubel, Jun 19 2021 *)
  • Sage
    def A156741(n, k): return round( product( binomial(2*(n+j), 2*(k+j))/binomial(2*(n-k+j), 2*j) for j in (0..8)) )
def A151709(n): return sum( A156741(n, k) for k in (0..n) )
[A151709(n) for n in (0..30)] # G. C. Greubel, Jun 19 2021

Formula

a(n) = Sum_{k=0..n} A156741(n, k).

Extensions

Terms a(11) onward added by G. C. Greubel, Jun 19 2021

A156740 Triangle T(n, k, m) = round( Product_{j=0..m} binomial(2*(n+j), 2*(k+j))/binomial( 2*(n-k+j), 2*j) ), where m = 7, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 153, 1, 1, 4845, 4845, 1, 1, 74613, 2362745, 74613, 1, 1, 735471, 358664691, 358664691, 735471, 1, 1, 5311735, 25533510145, 393216056233, 25533510145, 5311735, 1, 1, 30421755, 1056158828725, 160324910200455, 160324910200455, 1056158828725, 30421755, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 14 2009

Keywords

Examples

			Triangle begins as:
  1;
  1,       1;
  1,     153,           1;
  1,    4845,        4845,            1;
  1,   74613,     2362745,        74613,           1;
  1,  735471,   358664691,    358664691,      735471,       1;
  1, 5311735, 25533510145, 393216056233, 25533510145, 5311735, 1;

Links

Crossrefs

Cf. A086645 (m=0), A156739 (m=6), this sequence (m=7), A156741 (m=8), A156742 (m=9).
Cf. A151614 (row sums).

Programs

  • Magma
    A156740:= func< n,k | Round( (&*[Binomial(2*(n+j), 2*(k+j))/Binomial(2*(n-k+j), 2*j): j in [0..7]]) ) >;
[A156740(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 19 2021
  • Mathematica
    b[n_, k_]:= Binomial[2*n, 2*k];
T[n_, k_, m_]:= Round[Product[b[n+j, k+j]/b[n-k+j, j], {j,0,m}]];
Table[T[n, k, 7], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 19 2021 *)
  • Sage
    def A156740(n, k): return round( product( binomial(2*(n+j), 2*(k+j))/binomial(2*(n-k+j), 2*j) for j in (0..7)) )
flatten([[A156740(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 19 2021

Formula

T(n, k, m) = round( Product_{j=0..m} b(n+j, k+j)/b(n-k+j, j) ), where b(n, k) = binomial(2*n, 2*k) and m = 7.
Sum_{k=0..n} T(n, k, 7) = A151614(n).

Extensions

Definition corrected to give integral terms and edited by G. C. Greubel, Jun 19 2021

A156739 Triangle T(n, k, m) = round( Product_{j=0..m} binomial(2*(n+j), 2*(k+j))/binomial( 2*(n-k+j), 2*j) ), where m = 6, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 120, 1, 1, 3060, 3060, 1, 1, 38760, 988380, 38760, 1, 1, 319770, 103285710, 103285710, 319770, 1, 1, 1961256, 5226256926, 66199254396, 5226256926, 1961256, 1, 1, 9657700, 157843517260, 16494647553670, 16494647553670, 157843517260, 9657700, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 14 2009

Keywords

Examples

			Triangle begins as:
  1;
  1,       1;
  1,     120,          1;
  1,    3060,       3060,           1;
  1,   38760,     988380,       38760,          1;
  1,  319770,  103285710,   103285710,     319770,       1;
  1, 1961256, 5226256926, 66199254396, 5226256926, 1961256, 1;

Links

Crossrefs

Cf. A086645 (m=0), this sequence (m=6), A156740 (m=7), A156741 (m=8), A156742 (m=9).

Programs

  • Magma
    A156739:= func< n,k | Round( (&*[Binomial(2*(n+j), 2*(k+j))/Binomial(2*(n-k+j), 2*j): j in [0..6]]) ) >;
[A156739(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 18 2021
  • Mathematica
    b[n_, k_]:= Binomial[2*n, 2*k];
T[n_, k_, m_]:= Round[Product[b[n+j, k+j]/b[n-k+j, j], {j,0,m}]];
Table[T[n, k, 6], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 18 2021 *)
  • Sage
    def A156739(n, k): return round( product( binomial(2*(n+j), 2*(k+j))/binomial(2*(n-k+j), 2*j) for j in (0..6)) )
flatten([[A156739(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 18 2021

Formula

T(n, k, m) = round( Product_{j=0..m} b(n+j, k+j)/b(n-k+j, j) ), where b(n, k) = binomial(2*n, 2*k) and m = 6.

Extensions

Definition corrected to give integral terms and edited by G. C. Greubel, Jun 18 2021

A156742 Triangle T(n, k, m) = round( Product_{j=0..m} binomial(2*(n+j), 2*(k+j))/binomial( 2*(n-k+j), 2*j) ), where m = 9, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 231, 1, 1, 10626, 10626, 1, 1, 230230, 10590580, 230230, 1, 1, 3108105, 3097744650, 3097744650, 3108105, 1, 1, 30045015, 404255676825, 8758872997875, 404255676825, 30045015, 1, 1, 225792840, 29367745734600, 8590065627370500, 8590065627370500, 29367745734600, 225792840, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 14 2009

Keywords

Examples

			Triangle begins as:
  1;
  1,        1;
  1,      231,            1;
  1,    10626,        10626,             1;
  1,   230230,     10590580,        230230,            1;
  1,  3108105,   3097744650,    3097744650,      3108105,        1;
  1, 30045015, 404255676825, 8758872997875, 404255676825, 30045015, 1;

Links

Crossrefs

Cf. A086645 (m=0), A156739 (m=6), A156740 (m=7), A156741 (m=8), this sequence (m=9).

Programs

  • Magma
    A156742:= func< n,k | Round( (&*[Binomial(2*(n+j), 2*(k+j))/Binomial(2*(n-k+j), 2*j): j in [0..9]]) ) >;
[A156742(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 19 2021
  • Mathematica
    T[n_, k_, m_]:= Round[Product[Binomial[2*(n+j), 2*(k+j)]/Binomial[2*(n-k+j), 2*j], {j,0,m}]];
Table[T[n, k, 9], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 19 2021 *)
  • Sage
    def A156742(n, k): return round( product( binomial(2*(n+j), 2*(k+j))/binomial(2*(n-k+j), 2*j) for j in (0..9)) )
flatten([[A156742(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 19 2021

Formula

T(n, k, m) = round( Product_{j=0..m} b(n+j, k+j)/b(n-k+j, j) ), where b(n, k) = binomial(2*n, 2*k) and m = 9.

Extensions

Definition corrected to give integral terms and edited by G. C. Greubel, Jun 19 2021
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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