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 1-9 of 9 results.

A255961 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->j*k).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 6, 0, 1, 4, 12, 18, 13, 0, 1, 5, 18, 37, 47, 24, 0, 1, 6, 25, 64, 111, 110, 48, 0, 1, 7, 33, 100, 215, 303, 258, 86, 0, 1, 8, 42, 146, 370, 660, 804, 568, 160, 0, 1, 9, 52, 203, 588, 1251, 1938, 2022, 1237, 282, 0
Offset: 0

Views

Author

Alois P. Heinz, Mar 11 2015

Keywords

Comments

A(n,k) is the number of partitions of n when parts i are of k*i kinds. A(2,2) = 7: [2a], [2b], [2c], [2d], [1a,1a], [1a,1b], [1b,1b].

Examples

			Square array A(n,k) begins:
  1,  1,   1,    1,    1,     1,     1,     1, ...
  0,  1,   2,    3,    4,     5,     6,     7, ...
  0,  3,   7,   12,   18,    25,    33,    42, ...
  0,  6,  18,   37,   64,   100,   146,   203, ...
  0, 13,  47,  111,  215,   370,   588,   882, ...
  0, 24, 110,  303,  660,  1251,  2160,  3486, ...
  0, 48, 258,  804, 1938,  4005,  7459, 12880, ...
  0, 86, 568, 2022, 5400, 12150, 24354, 44885, ...

Links

Crossrefs

Columns k=0-10 give: A000007, A000219, A161870, A255610, A255611, A255612, A255613, A255614, A193427, A316461, A316462.
Rows n=0-3 give: A000012, A001477, A055998, A101853.
Main diagonal gives A255672.
Antidiagonal sums give A299166.
Cf. A144064, A257673.

Programs

  • Maple
    A:= proc(n, k) option remember; `if`(n=0, 1, k*add(
      A(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);
  • Mathematica
    A[n_, k_] := A[n, k] = If[n==0, 1, k*Sum[A[n-j, k]*DivisorSigma[2, j], {j, 1, n}]/n]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 02 2016, after Alois P. Heinz *)

Formula

G.f. of column k: Product_{j>=1} 1/(1-x^j)^(j*k).
T(n,k) = Sum_{i=0..k} C(k,i) * A257673(n,k-i).

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).

A299167 Expansion of 1/(1 - x*Product_{k>=1} (1 + x^k)^k).

Original entry on oeis.org

1, 1, 2, 5, 14, 36, 94, 243, 628, 1619, 4178, 10776, 27793, 71682, 184879, 476832, 1229830, 3171942, 8180989, 21100215, 54421187, 140361900, 362018270, 933709453, 2408202606, 6211182512, 16019743522, 41317765457, 106565859669, 274852289679, 708892898170, 1828360759013, 4715667307920
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A277938.
Cf. A026007, A067687, A299105, A299106, A299108, A299162, A299164, A299166.

Programs

  • Mathematica
    nmax = 32; 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} A026007(k-1)*a(n-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).
Showing 1-9 of 9 results.

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

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