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 1-6 of 6 results.

A318770 Expansion of Sum_{k>=0} x^(k^2) / Product_{j=1..k} (1 - j*x^j).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 4, 8, 9, 17, 19, 38, 42, 80, 97, 174, 208, 389, 460, 826, 1049, 1790, 2248, 3989, 4933, 8451, 11116, 18300, 23742, 40446, 51774, 85774, 115454, 184806, 245967, 406768, 533210, 860295, 1179570, 1850325, 2505585, 4046594, 5407269, 8556317, 11877833, 18327723
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 03 2018

Keywords

Crossrefs

Cf. A193196, A204856, A318771.

Programs

  • Maple
    a:=series(add(x^(k^2)/mul((1-j*x^j),j=1..k),k=0..100),x=0,47): seq(coeff(a,x,n),n=0..46); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 46; CoefficientList[Series[Sum[x^k^2/Product[(1 - j x^j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

A246565 Number of rooted ordered trees with n non-root nodes such that the branch lengths are weakly increasing.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 20, 31, 67, 115, 238, 406, 880, 1494, 3118, 5568, 11408, 20092, 41938, 73687, 151529, 272359, 552611, 987277, 2033173, 3617167, 7371745, 13318869, 26991289, 48496985, 99264686, 177917588, 362349258, 655925568, 1331038563, 2401097768, 4913906801, 8844673793, 18046697901, 32720071992, 66666578597
Offset: 0

Views

Author

Joerg Arndt, Aug 30 2014

Keywords

Examples

			The a(4) = 6 such trees are:
:
:   1:
:  [ 1 1 1 1 ]
:  [ 0 0 0 0 ]
:
:  O--o
:  .--o
:  .--o
:  .--o
:
:
:   2:
:  [ 1 1 2 ]
:  [ 0 0 0 ]
:
:  O--o
:  .--o
:  .--o--o
:
:
:   3:
:  [ 1 3 ]
:  [ 0 0 ]
:
:  O--o
:  .--o--o--o
:
:
:   4:
:  [ 2 2 ]
:  [ 0 0 ]
:
:  O--o--o
:  .--o--o
:
:
:   5:
:  [ 2 2 ]
:  [ 0 1 ]
:
:  O--o--o
:     .--o--o
:
:
:   6:
:  [ 4 ]
:  [ 0 ]
:
:  O--o--o--o--o
:
See the Arndt link for all trees for 0 <= n <= 7.

Links

Crossrefs

Cf. A000108 (all trees), A246566 (descending branch lengths), A193196, A192243.

A246566 Number of rooted ordered trees with n non-root nodes such that the branch lengths are weakly decreasing.

Original entry on oeis.org

1, 1, 2, 4, 10, 24, 62, 155, 396, 992, 2504, 6228, 15512, 38220, 94059, 229498, 558642, 1350384, 3255138, 7801327, 18642118, 44329833, 105111283, 248184255, 584396849, 1371048495, 3208318247, 7483522231, 17413665881, 40405120865, 93543352492
Offset: 0

Views

Author

Joerg Arndt, Aug 30 2014

Keywords

Examples

			The a(3) = 4 such trees are:
:
:     1:
:    [ 1 1 1 ]
:    [ 0 0 0 ]
:
:  O--o
:  .--o
:  .--o
:
:
:     2:
:    [ 2 1 ]
:    [ 0 0 ]
:
:  O--o--o
:  .--o
:
:
:     3:
:    [ 2 1 ]
:    [ 0 1 ]
:
:  O--o--o
:     .--o
:
:
:     4:
:    [ 3 ]
:    [ 0 ]
:
:  O--o--o--o
:
See the Arndt link for all trees for 0 <= n <= 5.

Links

Crossrefs

Cf. A000108 (all trees), A246565 (ascending branch lengths), A193196, A192243, A001519.

A306702 Expansion of Sum_{k>=0} x^k / Product_{j=1..k} (1 + j*x^j).

Original entry on oeis.org

1, 1, 0, 1, -2, 1, -1, 1, -7, 10, 0, 4, -31, 31, 8, 53, -163, 108, -74, 212, -450, 732, -353, 467, -3412, 3614, -145, 5613, -11910, 8816, -13354, 21558, -44624, 77598, -43860, 67721, -255791, 261710, -127452, 529648, -1118393, 997295, -1206756, 2184148, -3314638, 5934992, -5394856
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 05 2019

Keywords

Crossrefs

Cf. A081362, A193196.

Programs

  • Mathematica
    nmax = 46; CoefficientList[Series[Sum[x^k/Product[(1 + j x^j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

A306663 Expansion of Sum_{k>=0} x^(k*(k+1)) / Product_{j=1..k} (1 - j*x^j).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 2, 4, 4, 8, 8, 17, 17, 35, 38, 74, 80, 161, 173, 336, 387, 713, 818, 1555, 1765, 3248, 3923, 6905, 8282, 15012, 17814, 31419, 39321, 66679, 82923, 144789, 177721, 302789, 390123, 642640, 821316, 1390825, 1755400, 2910638, 3833338, 6165743, 8060128, 13322378
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 04 2019

Keywords

Crossrefs

Cf. A003106, A193196, A204856.

Programs

  • Mathematica
    nmax = 48; CoefficientList[Series[Sum[x^(k (k + 1))/Product[(1 - j x^j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

A306730 Expansion of Sum_{k>=0} x^k * Product_{j=1..k} (1 + j*x^j).

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 9, 12, 22, 32, 45, 70, 98, 153, 231, 318, 452, 685, 933, 1387, 1988, 2769, 3850, 5433, 7622, 10530, 14896, 20635, 28123, 38934, 53067, 72144, 99509, 133661, 183794, 249130, 333956, 448306, 605165, 807053, 1078093, 1444021, 1913418, 2556415, 3406636
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 06 2019

Keywords

Crossrefs

Cf. A000009, A193196, A306702.

Programs

  • Mathematica
    nmax = 44; CoefficientList[Series[Sum[x^k Product[(1 + j x^j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
Showing 1-6 of 6 results.

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