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 1-10 of 10 results.

A299208 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + x^k)).

Original entry on oeis.org

1, 1, 0, -1, -2, -1, 1, 3, 3, 1, -3, -6, -5, 1, 9, 12, 5, -9, -20, -18, 1, 26, 38, 21, -21, -61, -62, -9, 72, 120, 81, -44, -177, -205, -64, 186, 366, 293, -63, -496, -657, -304, 445, 1084, 1014, 33, -1341, -2053, -1238, 959, 3132, 3378, 770, -3474, -6260, -4619, 1656, 8809, 10929, 4306, -8520
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 05 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A286352.
Cf. similar sequences: A067687, A299105, A299106, A299208, A302017, A318581, A318582, A331484.
Cf. A081362, A299108, A299162, A299164, A299166, A299167, A299209, A299210, A299211, A299212.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[1/(1 - x Product[1/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} 1/(1 + x^k)).
a(0) = 1; a(n) = Sum_{k=1..n} A081362(k-1)*a(n-k).

A299162 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 - k*x^k)).

Original entry on oeis.org

1, 1, 2, 6, 17, 49, 135, 380, 1051, 2925, 8119, 22548, 62574, 173767, 482360, 1339126, 3717700, 10321163, 28653557, 79548612, 220843925, 613110573, 1702128034, 4725475979, 13118945083, 36421037100, 101112695940, 280710759278, 779313926949, 2163544401343, 6006468273440
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A297328.
Cf. A006906, A067687, A299105, A299106, A299108, A299164, A299166, A299167.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[1/(1 - x Product[1/(1 - k x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} 1/(1 - k*x^k)).
a(0) = 1; a(n) = Sum_{k=1..n} A006906(k-1)*a(n-k).

A299164 Expansion of 1/(1 - x*Product_{k>=1} (1 + k*x^k)).

Original entry on oeis.org

1, 1, 2, 5, 14, 35, 91, 233, 597, 1517, 3885, 9922, 25333, 64683, 165181, 421828, 1077277, 2750993, 7025168, 17940298, 45814165, 116996152, 298774246, 762982615, 1948434235, 4975732669, 12706571546, 32448880807, 82864981016, 211613009498, 540397935771, 1380018797044, 3524165721799
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A297321.
Cf. A022629, A067687, A299105, A299106, A299108, A299162, A299166, A299167.

Programs

  • Mathematica
    nmax = 32; CoefficientList[Series[1/(1 - x Product[1 + k x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} (1 + k*x^k)).
a(0) = 1; a(n) = Sum_{k=1..n} A022629(k-1)*a(n-k).

A299166 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 - x^k)^k).

Original entry on oeis.org

1, 1, 2, 6, 17, 48, 132, 365, 1003, 2759, 7583, 20843, 57283, 157442, 432719, 1189317, 3268818, 8984318, 24693343, 67869557, 186539251, 512702559, 1409161449, 3873076007, 10645137706, 29258128633, 80415877302, 221022792843, 607480469466, 1669658209311, 4589050472041
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A255961.
Cf. A000219, A067687, A299105, A299106, A299108, A299162, A299164, A299167.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(n=0, 1, k*add(
       b(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
    end:
a:= n-> add(b(n-j, j), j=0..n):
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 04 2018
  • Mathematica
    nmax = 30; CoefficientList[Series[1/(1 - x Product[1/(1 - x^k)^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} 1/(1 - x^k)^k).
a(0) = 1; a(n) = Sum_{k=1..n} A000219(k-1)*a(n-k).

A277938 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1+x^j)^(j*k) in powers of x.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 5, 0, 1, 4, 9, 14, 8, 0, 1, 5, 14, 28, 30, 16, 0, 1, 6, 20, 48, 72, 68, 28, 0, 1, 7, 27, 75, 141, 183, 145, 49, 0, 1, 8, 35, 110, 245, 396, 443, 298, 83, 0, 1, 9, 44, 154, 393, 751, 1058, 1026, 600, 142, 0, 1, 10, 54, 208
Offset: 0

Views

Author

Seiichi Manyama, Apr 11 2017

Keywords

Examples

			Square array begins:
   1, 1,  1,  1,   1, ...
   0, 1,  2,  3,   4, ...
   0, 2,  5,  9,  14, ...
   0, 5, 14, 28,  48, ...
   0, 8, 30, 72, 141, ...

Links

Crossrefs

Columns k=0-4 give: A000007, A026007, A026011, A027346, A027906.
Rows n=0-3 give: A000012, A001477, A000096, A005586.
Main diagonal gives A270922.
Antidiagonal sums give A299167.
Cf. A276554, A279928.

Formula

G.f. of column k: Product_{j>=1} (1+x^j)^(j*k).

A299211 Expansion of 1/(1 - x*Product_{k>=1} (1 - x^k)^k).

Original entry on oeis.org

1, 1, 0, -3, -6, -4, 12, 39, 52, -9, -186, -392, -285, 610, 2291, 3200, -150, -10626, -23487, -18841, 32957, 134848, 198246, 13961, -605248, -1409604, -1234474, 1744213, 7898753, 12209679, 2161666, -34344627, -84393284, -79993042, 90692470, 461463974, 749309529, 207447895, -1939084232
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 05 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A276554.
Cf. A067687, A073592, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299209, A299210, A299212.

Programs

  • Maple
    N:= 100: # for a(0)..a(N)
S:= series(1/(1-x*mul((1-x^k)^k,k=1..N)),x,N+1):
seq(coeff(S,x,i),i=0..N); # Robert Israel, Feb 05 2023
  • Mathematica
    nmax = 38; CoefficientList[Series[1/(1 - x Product[(1 - x^k)^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} (1 - x^k)^k).
a(0) = 1; a(n) = Sum_{k=1..n} A073592(k-1)*a(n-k).

A299209 Expansion of 1/(1 - x*Product_{k>=1} (1 - k*x^k)).

Original entry on oeis.org

1, 1, 0, -3, -6, -5, 11, 37, 59, 13, -155, -402, -415, 263, 1981, 3748, 2289, -6643, -22642, -31322, -187, 99040, 229410, 216823, -230029, -1223267, -2097812, -955237, 4468902, 13393758, 16752461, -3891704, -62382597, -131974181, -106680562, 173622424, 741553622, 1163057561, 329176545
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 05 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A297323.
Cf. A022661, A067687, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299210, A299211, A299212.

Programs

  • Mathematica
    nmax = 38; CoefficientList[Series[1/(1 - x Product[1 - k x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} (1 - k*x^k)).
a(0) = 1; a(n) = Sum_{k=1..n} A022661(k-1)*a(n-k).

A299210 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + k*x^k)).

Original entry on oeis.org

1, 1, 0, -2, -5, -3, 5, 20, 27, 17, -53, -152, -192, 31, 576, 1110, 694, -1297, -4519, -6160, -1107, 13665, 31914, 30643, -19339, -119260, -196142, -103318, 289543, 859631, 1062684, 13710, -2690348, -5675946, -4940757, 4167527, 21343918, 33874107, 16524162, -51704908, -150454546
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 05 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A297325.
Cf. A022693, A067687, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299209, A299211, A299212.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[1/(1 - x Product[1/(1 + k x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} 1/(1 + k*x^k)).
a(0) = 1; a(n) = Sum_{k=1..n} A022693(k-1)*a(n-k).

A299212 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + x^k)^k).

Original entry on oeis.org

1, 1, 0, -2, -5, -4, 4, 21, 35, 23, -47, -165, -239, -78, 479, 1273, 1508, -138, -4429, -9451, -8845, 6207, 37937, 67123, 45144, -83355, -308078, -455109, -166872, 873799, 2393041, 2916869, -73472, -8133572, -17828640, -17294146, 10383571, 70275162, 127401305, 90368779, -147825714
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 05 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A279928.
Cf. A067687, A255528, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299209, A299210, A299211.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[1/(1 - x Product[1/(1 + x^k)^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} 1/(1 + x^k)^k).
a(0) = 1; a(n) = Sum_{k=1..n} A255528(k-1)*a(n-k).

A307062 Expansion of 1/(2 - Product_{k>=1} (1 + x^k)^k).

Original entry on oeis.org

1, 1, 3, 10, 29, 88, 264, 790, 2366, 7086, 21216, 63523, 190201, 569485, 1705121, 5105383, 15286247, 45769238, 137039743, 410316854, 1228548190, 3678451550, 11013817655, 32976968175, 98737827756, 295635383297, 885175234817, 2650343093602, 7935511791620, 23760073760720, 71141108467679
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 21 2019

Keywords

Comments

Invert transform of A026007.
a(n) is the number of compositions of n where there are A026007(k) sorts of part k. - Joerg Arndt, Jan 24 2024

Links

Crossrefs

Cf. A026007, A257674, A299167, A304969.
Cf. A307057, A307058, A307059, A307060, A307063.

Programs

  • Magma
    m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(2 - (&*[(1+x^j)^j: j in [1..m+2]])) )); // G. C. Greubel, Jan 24 2024
  • Maple
    b:= proc(n) b(n):= add((-1)^(n/d+1)*d^2, d=numtheory[divisors](n)) end:
g:= proc(n) g(n):= `if`(n=0, 1, add(b(k)*g(n-k), k=1..n)/n) end:
a:= proc(n) a(n):= `if`(n=0, 1, add(g(k)*a(n-k), k=1..n)) end:
seq(a(n), n=0..45);  # Alois P. Heinz, Jan 24 2024
  • Mathematica
    nmax = 30; CoefficientList[Series[1/(2 - Product[(1 + x^k)^k, {k, 1, nmax}]), {x, 0, nmax}], x]
  • SageMath
    m=80;
def f(x): return 1/( 2 - product((1+x^j)^j for j in range(1,m+3)) )
def A307062_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307062_list(m) # G. C. Greubel, Jan 24 2024

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A026007(k)*a(n-k).
Showing 1-10 of 10 results.

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