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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A062534 Table by antidiagonals of coefficient of x^k in expansion of 1/((1+x)^2*(1-x)^n).

Original entry on oeis.org

1, -2, 1, 3, -1, 1, -4, 2, 0, 1, 5, -2, 2, 1, 1, -6, 3, 0, 3, 2, 1, 7, -3, 3, 3, 5, 3, 1, -8, 4, 0, 6, 8, 8, 4, 1, 9, -4, 4, 6, 14, 16, 12, 5, 1, -10, 5, 0, 10, 20, 30, 28, 17, 6, 1, 11, -5, 5, 10, 30, 50, 58, 45, 23, 7, 1, -12, 6, 0, 15, 40, 80, 108, 103, 68, 30, 8, 1, 13, -6, 6, 15, 55, 120, 188, 211, 171, 98, 38, 9, 1, -14, 7, 0, 21, 70, 175
Offset: 0

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Author

Henry Bottomley, Jun 25 2001

Keywords

Crossrefs

Rows are effectively (with minor adjustments): A038608, A001057, A027656, A008805, A006918, A002624, A028346. Cf. A058394 which (adjusting for signs and an overlap of three rows) is effectively the continuation of this table for negative n.

Formula

Each row is partial sum of preceding row, i.e. T(n, k)=T(n-1, k)+T(n, k-1) with T(0, k)=(k+1)*(-1)^k and T(n, 0)=1.

A135247 a(n) = 3*a(n-1) + 2*a(n-2) - 6*a(n-3).

Original entry on oeis.org

1, 3, 11, 33, 103, 309, 935, 2805, 8431, 25293, 75911, 227733, 683263, 2049789, 6149495, 18448485, 55345711, 166037133, 498111911, 1494335733, 4483008223, 13449024669, 40347076055, 121041228165, 363123688591, 1089371065773, 3268113205511, 9804339616533
Offset: 0

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Author

Paul Curtz, Feb 15 2008

Keywords

Comments

This sequence interleaves A016133 and 3*A016133, see formulas. - Mathew Englander, Jan 08 2024
a(n) is the number of partitions of n into parts 1 (in three colors) and 2 (in two colors) where the order of colors matters. For example, the a(2)=11 such partitions (using parts 1, 1', 1'', 2, and 2') are 2, 2', 1+1, 1+1', 1+1'', 1'+1, 1'+1', 1'+1'', 1''+1, 1''+1', 1''+1''. For such partitions where the order of colors does not matter see A002624. - Joerg Arndt, Jan 18 2024

Links

Crossrefs

Cf. A016133.

Programs

  • GAP
    a:=[1,3,11];; for n in [4..30] do a[n]:=3*a[n-1]+2*a[n-2]-6*a[n-3]; od; a; # G. C. Greubel, Nov 20 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1-3*x-2*x^2+6*x^3) )); // G. C. Greubel, Nov 20 2019
  • Maple
    seq(coeff(series(1/(1-3*x-2*x^2+6*x^3), x, n+1), x, n), n = 0..30); # G. C. Greubel, Nov 20 2019
  • Mathematica
    LinearRecurrence[{3,2,-6},{1,3,11},30] (* Harvey P. Dale, Jun 27 2015 *)
  • PARI
    my(x='x+O('x^30)); Vec(1/(1-3*x-2*x^2+6*x^3)) \\ G. C. Greubel, Nov 20 2019
  • Sage
    def A135247_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-3*x-2*x^2+6*x^3) ).list()
A135247_list(30) # G. C. Greubel, Nov 20 2019

Formula

G.f.: 1/((1-3*x)*(1-2*x^2)). - G. C. Greubel, Oct 04 2016
From Mathew Englander, Jan 08 2024: (Start)
a(n) = A010684(n) * A016133(floor(n/2)).
a(n) = 3*a(n-1) + A077957(n) for n >= 1.
a(n) = (A000244(n+2) - A164073(n+3))/7.
(End)

Extensions

More terms from Harvey P. Dale, Jun 27 2015
Dropped two leading terms = 0. - Joerg Arndt, Jan 18 2024

A287143 Expansion of x*(1 + 3*x + x^2)/((1 - x)^5*(1 + x)^4).

Original entry on oeis.org

0, 1, 4, 9, 21, 35, 65, 95, 155, 210, 315, 406, 574, 714, 966, 1170, 1530, 1815, 2310, 2695, 3355, 3861, 4719, 5369, 6461, 7280, 8645, 9660, 11340, 12580, 14620, 16116, 18564, 20349, 23256, 25365, 28785, 31255, 35245, 38115, 42735, 46046, 51359, 55154, 61226, 65550, 72450, 77350, 85150, 90675, 99450, 105651, 115479
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 15 2017

Keywords

Links

Crossrefs

Cf. A002413, A002418, A002419, A002624, A027656, A085787, A212246, A290055.

Programs

  • Mathematica
    CoefficientList[Series[x (1 + 3 x + x^2)/((1 - x)^5 (1 + x)^4), {x, 0, 52}], x]
LinearRecurrence[{1, 4, -4, -6, 6, 4, -4, -1, 1}, {0, 1, 4, 9, 21, 35, 65, 95, 155}, 53]

Formula

G.f.: x*(1 + 3*x + x^2)/((1 - x)^5*(1 + x)^4).
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9).
Generalized 4-dimensional figurate numbers (A002418): (5*n - 1)*binomial(n + 2,3)/4, n = 0,+1,-3,+2,-4,+3,-5, ...
Convolution of the sequences A027656 and A085787.
a(n) = (2*n+3+(-1)^n)*(2*n+7+(-1)^n)*(5*(2*n^2+10*n+3)-3*(2*n+5)*(-1)^n)/3072. - Luce ETIENNE, Nov 18 2017

A290055 Expansion of x*(1 + 4*x + x^2)/((1 - x)^5*(1 + x)^4).

Original entry on oeis.org

0, 1, 5, 10, 26, 40, 80, 110, 190, 245, 385, 476, 700, 840, 1176, 1380, 1860, 2145, 2805, 3190, 4070, 4576, 5720, 6370, 7826, 8645, 10465, 11480, 13720, 14960, 17680, 19176, 22440, 24225, 28101, 30210, 34770, 37240, 42560, 45430, 51590, 54901, 61985, 65780, 73876, 78200, 87400, 92300, 102700, 108225, 119925, 126126
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 15 2017

Keywords

Comments

More generally, the generalized 4-dimensional figurate numbers are convolution of the sequence {1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...} with generalized polygonal numbers (A195152).

Links

Crossrefs

Cf. A001082, A002414, A002419, A002624, A027656, A195152, A287143.

Programs

  • Mathematica
    CoefficientList[Series[x (1 + 4 x + x^2)/((1 - x)^5 (1 + x)^4), {x, 0, 51}], x]
LinearRecurrence[{1, 4, -4, -6, 6, 4, -4, -1, 1}, {0, 1, 5, 10, 26, 40, 80, 110, 190}, 52]
  • PARI
    x='x+O('x^99); concat(0, Vec(x*(1+4*x+x^2)/((1-x)^5*(1 + x)^4))) \\ Altug Alkan, Aug 15 2017

Formula

G.f.: x*(1 + 4*x + x^2)/((1 - x)^5*(1 + x)^4).
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9).
Generalized 4-dimensional figurate numbers (A002419): (3*n - 1)*binomial(n + 2,3)/2, n = 0,+1,-3,+2,-4,+3,-5, ...
Convolution of the sequences A027656 and A001082 (with offset 0).
a(n) = (2*n+3+(-1)^n)*(2*n+7+(-1)^n)*(6*n^2+30*n+5-(2*n+5)*(-1)^n)/1536. - Luce ETIENNE, Nov 18 2017

A259325 Infinite square array T(n,k) read by antidiagonals, defined by T(n,k) = T(n,k-1)+T(n-k,k), T(0,k)=1 (n >= 0, k >= 1).

Original entry on oeis.org

1, 1, 3, 1, 3, 7, 1, 3, 8, 13, 1, 3, 8, 16, 22, 1, 3, 8, 17, 30, 34, 1, 3, 8, 17, 33, 50, 50, 1, 3, 8, 17, 34, 58, 80, 70, 1, 3, 8, 17, 34, 61, 97, 120, 95
Offset: 0

Views

Author

N. J. A. Sloane, Jun 24 2015

Keywords

Examples

			The first few antidiagonals are:
1
1,3,
1,3,7
1,3,8,13
1,3,8,16,22
1,3,8,17,30,34
1,3,8,17,33,50,50
1,3,8,17,34,58,80,70
1,3,8,17,34,61,97,120,95
...

Links

Crossrefs

Columns give A002623, A002624, A002625, A002626.
Previous Showing 11-15 of 15 results.

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

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