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 1-5 of 5 results.

A327000 A(n, k) = A309522(n, k) - A327001(n, k) for n >= 0 and k >= 3, square array read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 6, 3, 9, 26, 10, 117, 68, 100, 35, 2574, 4500, 517, 365, 126, 70005, 748616, 199155, 4163, 1302, 462, 2082759, 192426260, 282846568, 10499643, 36180, 4606, 1716, 65061234, 59688349943, 799156187475, 141482705378, 663488532, 341733, 16284
Offset: 0

Views

Author

Peter Luschny, Aug 12 2019

Keywords

Examples

			Array starts:
n\k [  3    4        5            6                 7 ]
[0]    1,   6,       26,          100,              365, ...            [A125107]
[1]    1,   9,       68,          517,              4163, ...           [A048742]
[2]    3,   117,     4500,        199155,           10499643, ...       [A326995]
[3]    10,  2574,    748616,      282846568,        141482705378, ...   [A327002]
[4]    35,  70005,   192426260,   799156187475,     4961959681629275, ...
[5]    126, 2082759, 59688349943, 3097220486457142, 278271624962638244163, ...
   A001700,

Crossrefs

Cf. A327001, A309522, A125107, A048742, A001700, A291973, A326999, A326995, A327002.

Programs

  • Maple
    ListTools:-Flatten([seq(seq(A309522(n-k, k) - A327001(n-k, k), k=3..n), n=3..10)]);

Formula

The columns for k = 0, 1, 2 are suppressed as they are identical 0.
A(0, k) = A000108(k) - A011782(k).
A(1, k) = A000142(k) - A000110(k).
A(2, k) = A002105(k) - A005046(k-1) for k >= 1.
A(3, k) = A018893(k) - A291973(k).
A(4, k) = A326999(k) - A291975(k).

A326997 Antidiagonal sums of A327001.

Original entry on oeis.org

1, 2, 4, 8, 19, 75, 866, 38546, 6435801, 4411898775, 12497557933746, 172551409850674506, 10689931902245247171663, 3609784804719064481553949028, 6092547417568499138450938430843772, 57094466457110460366185443558858325798634
Offset: 0

Views

Author

Peter Luschny, Aug 13 2019

Keywords

Programs

  • Maple
    seq(add(A327001(n-k, k), k=0..n), n=0..15);

A326996 Main diagonal of A260876.

Original entry on oeis.org

1, 1, 4, 365, 3086331, 5612271711927, 3829797188212731601783, 1478967550753025951356611840021161, 452501475882033837823819972299248189399008553836, 146630849220097180917463638217405949960396188742877031073909770851
Offset: 0

Views

Author

Peter Luschny, Aug 13 2019

Keywords

Links

Crossrefs

Cf. A260876, A327001.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(n=0, 1, add(
       binomial(k*n-1, k*j-1)*b(n-j, k), j=1..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..10);  # Alois P. Heinz, Aug 14 2019
  • Mathematica
    b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[Binomial[k n - 1, k j - 1] b[n - j, k], {j, 1, n}]];
a[n_] := b[n, n];
a /@ Range[0, 10] (* Jean-François Alcover, Nov 07 2020, after Maple *)

Formula

a(n) = A260876(n,n).
a(n) = (n^2)! * [x^(n^2)] exp(Sum_{k>=1} x^(n*k) / (n*k)!). - Ilya Gutkovskiy, Feb 09 2020

A326998 a(n) = 1 + binomial(3*n-1, n) + binomial(3*n-1, n-1)*(binomial(2*n-1, n) + 1).

Original entry on oeis.org

4, 5, 31, 365, 6271, 129130, 2877421, 66628441, 1578320767, 37983592076, 925196176906, 22754692780561, 564123212097901, 14079691134569845, 353428830512017081, 8915830309096530865, 225890912989184760703, 5744976464242932324976, 146603288011226858621356
Offset: 0

Views

Author

Peter Luschny, Aug 13 2019

Keywords

Crossrefs

Cf. A327001 (column 3). Essentially the same as A309725.

Programs

  • Maple
    a := n -> 1 + binomial(3*n-1, n) + binomial(3*n-1, n-1)*(binomial(2*n-1, n) + 1):
seq(a(n), n=0..19);

Formula

From Mike Sheppard, Feb 17 2025: (Start)
G.f.: (1/6) * (11 + 6/(1 - x) + (12*cos(1/6 arccos(1 - (27*x)/2)))/sqrt(4 - 27*x) + hypergeom([1/3, 2/3], [1], 27*x)).
E.g.f.: exp(x) + hypergeom([1/3, 2/3], [1/2, 1], (27*x)/4) + (1/6) * (11 + hypergeom([1/3, 2/3], [1, 1], 27*x)). (End)

A326999 a(n) = A327000(4, n) + A291975(n).

Original entry on oeis.org

1, 1, 36, 6306, 3156336, 3501788976, 7425169747776, 27322211071838736, 163058737794666253056, 1500955605765318574331136, 20488125782700503099836056576, 401537703770887804145153979250176, 10992280532048388256580758224034983936, 410332411533091221236570416481170032685056
Offset: 0

Views

Author

Peter Luschny, Aug 12 2019

Keywords

Crossrefs

Cf. A327000, A309522, A327001, A291975.

Formula

a(n) = A309522(4, n) - A327001(4, n) + A291975(n).
Showing 1-5 of 5 results.

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