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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A344101 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+5,6).

Original entry on oeis.org

1, 1, 7, 35, 133, 511, 1869, 6797, 24095, 83938, 286734, 964348, 3196984, 10460310, 33813984, 108076908, 341821250, 1070484009, 3321584021, 10217036263, 31169524988, 94351439060, 283498600776, 845848778722, 2506779443603, 7381617323598, 21603241378334, 62853440151768
Offset: 0

Views

Author

Ilya Gutkovskiy, May 09 2021

Keywords

Crossrefs

Cf. A000428, A000579, A028377, A258343, A305206, A344099, A344100.

Programs

  • Mathematica
    nmax = 27; CoefficientList[Series[Product[(1 + x^k)^Binomial[k + 5, 6], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d Binomial[d + 5, 6], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 27}]

Formula

G.f.: exp( Sum_{k>=1} (-1)^(k+1) * x^k / (k*(1 - x^k)^7) ).

A298850 Expansion of Product_{k>=1} (1 + x^(k*(k+1)/2))^(k*(k+1)/2).

Original entry on oeis.org

1, 1, 0, 3, 3, 0, 9, 9, 0, 19, 29, 10, 33, 63, 30, 66, 156, 90, 110, 300, 235, 276, 561, 465, 558, 1083, 1065, 1154, 1877, 1983, 2295, 3834, 3879, 3861, 6858, 7452, 7561, 12613, 13252, 13057, 22161, 25569, 24582, 35985, 44193, 44970, 63495, 79105, 77143, 104046, 134820, 138759, 182511, 222600
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 09 2018

Keywords

Crossrefs

Cf. A000217, A024940, A027999, A028377, A291649, A298730.

Programs

  • Mathematica
    nmax = 53; CoefficientList[Series[Product[(1 + x^(k (k + 1)/2))^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^A000217(k))^A000217(k).

A371481 Expansion of e.g.f. Product_{k>=1} (1 + x^k/k!)^(k*(k+1)/2).

Original entry on oeis.org

1, 1, 3, 15, 52, 335, 2031, 12880, 102614, 802443, 6956995, 64721756, 633754320, 6551212057, 71375943289, 820250162880, 9747543483676, 121775559807881, 1580353806494781, 21246545374234378, 296590230821338520, 4280692741624646151, 63852747607056438283
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 25 2024

Keywords

Comments

"EGJ" (unordered, element, labeled) transform of triangular numbers.

Links

Crossrefs

Cf. A000217, A007837, A028377, A032315, A032316, A371309.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Product[(1 + x^k/k!)^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Previous Showing 31-33 of 33 results.

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

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