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.

A223940 Sums of antidiagonals of A223968.

Original entry on oeis.org

1, 2, 4, 8, 16, 31, 61, 117, 228, 436, 845, 1615, 3120, 5965, 11501, 22001, 42365, 81091, 156010, 298777, 574450, 1100620, 2115150, 4053959, 7788126, 14931102, 28676899, 54990202, 105594073, 202519004, 388825095, 745825185, 1431776536, 2746639052
Offset: 0

Views

Author

Philippe Deléham, Mar 29 2013

Keywords

Links

Crossrefs

Cf. A223968

Programs

  • Mathematica
    CoefficientList[Series[(1-x)(1+2x-x^3)/(1-x-4x^2+3x^3+3x^4-x^5), {x,0,40}],x] (* or *) LinearRecurrence[{1,4,-3,-3,1},{1,2,4,8,16},40] (* Harvey P. Dale, Jul 04 2019 *)

Formula

G.f.: (1-x) * (1+2*x-x^3) / (1-x-4*x^2+3*x^3+3*x^4-x^5).
a(n) = a(n-1) + 4*a(n-2) - 3*a(n-3) - 3*a(n-4) + a(n-5) with a(0) = 1, a(1) = 2, a(2) = 4, a(3) = 8, a(4) = 16, a(5) = 31.
a(n) = Sum_{k=0..n} A223968(n-k, k).

Extensions

a(32) corrected by Sean A. Irvine, May 19 2019

A221863 Expansion of (1-3*x+x^2)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).

Original entry on oeis.org

1, 6, 27, 110, 429, 1637, 6172, 23104, 86090, 319792, 1185305, 4386331, 16212913, 59873834, 220964744, 815057639, 3005282745, 11077802256, 40824723483, 150424044413, 554183037617, 2041477665799, 7519722443381, 27696997721940, 102010147865915, 375697698147882
Offset: 0

Views

Author

Philippe Deléham, Apr 10 2013

Keywords

Comments

A diagonal of the square array A223968.

Links

Crossrefs

Cf. A223968

Programs

  • Mathematica
    CoefficientList[Series[(1-3x+x^2)/(1-9x+28x^2-35x^3+15x^4-x^5),{x,0,30}],x] (* or *) LinearRecurrence[{9,-28,35,-15,1},{1,6,27,110,429},30] (* Harvey P. Dale, Jan 26 2015 *)

Formula

a(n) = A223968(n,n+4) = A223968(n,n+5).
G.f.: (1-3*x+x^2)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).
a(n) = 9*a(n-1) - 28*a(n-2) + 35*a(n-3) - 15*a(n-4) + a(n-5) with a(0) = 1, a(1) = 6, a(2) = 27, a(3) = 110, a(4) = 429.

A224422 Expansion of (1-x)*(1-3*x)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).

Original entry on oeis.org

1, 5, 20, 75, 275, 1001, 3639, 13243, 48280, 176341, 645150, 2363596, 8669142, 31825005, 116914020, 429737220, 1580244061, 5812839156, 21387636101, 78708626396, 289699273501, 1066406842677, 3925882147566, 14453780545834, 53216783798234, 195944670698910
Offset: 0

Views

Author

Philippe Deléham, Apr 06 2013

Keywords

Comments

A diagonal of the square array A223968.

Links

Crossrefs

Cf. A223968.

Programs

  • Mathematica
    LinearRecurrence[{9, -28, 35, -15, 1}, {1, 5, 20, 75, 275}, 26] (* Michael De Vlieger, Aug 05 2021 *)

Formula

a(n) = A223968(n+3,n) = A223968(n+4,n).
a(n) = 9*a(n-1) - 28*a(n-2) + 35*a(n-3) - 15*a(n-4) + a(n-5) with a(0) = 1, a(1) = 5, a(2) = 20, a(3) = 75, a(4) = 275.

A216710 Expansion of (1-3*x+x^2)^2/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).

Original entry on oeis.org

1, 3, 10, 35, 126, 460, 1690, 6225, 22950, 84626, 312019, 1150208, 4239225, 15621426, 57556155, 212037241, 781074572, 2877011660, 10596599460, 39027676220, 143735627861, 529352597361, 1949472483601, 7179308057596, 26438877143476, 97364252272077
Offset: 0

Views

Author

Philippe Deléham, Apr 09 2013

Keywords

Comments

A diagonal of the square array A223968.

Links

Crossrefs

Cf. A223968.

Programs

  • Mathematica
    CoefficientList[Series[(1 - 3 x + x^2)^2/(1 - 9 x + 28 x^2 - 35 x^3 + 15 x^4 - x^5), {x, 0, 25}], x] (* Michael De Vlieger, Aug 19 2015 *)
LinearRecurrence[{9, -28, 35, -15, 1},{1, 3, 10, 35, 126},26] (* Ray Chandler, Aug 28 2015 *)
  • PARI
    Vec((1-3*x+x^2)^2/(1-9*x+28*x^2-35*x^3+15*x^4-x^5) + O(x^30)) \\ Colin Barker, Aug 19 2015

Formula

a(n) = A223968(n,n+1).
G.f.: (1-3*x+x^2)^2/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).
a(n) = 9*a(n-1) - 28*a(n-2) + 35*a(n-3) - 15*a(n-4) + a(n-5) with a(0) = 1, a(1) = 3, a(2) = 10, a(3) = 35, a(4) = 126.

A224514 Expansion of (1-x)*(1-3*x)*(1-3*x+x^2)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).

Original entry on oeis.org

1, 2, 6, 20, 70, 251, 911, 3327, 12190, 44744, 164407, 604487, 2223504, 8181175, 30108147, 110820165, 407946421, 1501844193, 5529362694, 20358557249, 74961030414, 276017648570, 1016360893036, 3742540945813, 13781324308298, 50748099850042
Offset: 0

Views

Author

Philippe Deléham, Apr 08 2013

Keywords

Comments

A diagonal of the square array A223968.

Links

Crossrefs

Cf. A223968

Programs

  • Mathematica
    LinearRecurrence[{9,-28,35,-15,1},{1,2,6,20,70,251,911,3327,12190,44744,164407,604487,2223504,8181175},40] (* Harvey P. Dale, Apr 24 2016 *)

Formula

a(n) = A223968(n,n).
G.f.: (1-x)*(1-3*x)*(1-3*x+x^2)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).
a(n) = 9*a(n-1) - 28*a(n-2) + 35*a(n-3) - 15*a(n-4) + a(n-5) with a(0) = 1, a(1) = 2, a(2) = 6, a(3) = 20, a(4) = 70.

Extensions

a(8) corrected by Georg Fischer, May 10 2019

A220948 Expansion of (1-x)^2*(1-3*x)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).

Original entry on oeis.org

1, 4, 15, 55, 200, 726, 2638, 9604, 35037, 128061, 468809, 1718446, 6305546, 23155863, 85089015, 312823200, 1150506841, 4232595095, 15574796945, 57320990295, 210990647105, 776707569176, 2859475304889, 10527898398268, 38763003252400, 142727886900676, 525547912974105
Offset: 0

Views

Author

Philippe Deléham, Apr 07 2013

Keywords

Comments

A diagonal of the square array A223968.

Links

Crossrefs

Cf. A223968

Formula

a(n) = A223968(n+2,n).
G.f.: (1-x)^2*(1-3*x)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).
a(n) = 9*a(n-1) - 28*a(n-2) + 35*a(n-3) - 15*a(n-4) + a(n-5) with a(0) = 1, a(1) = 4, a(2) = 15, a(3) = 55, a(4) = 200.

A221859 Expansion of (1-3*x+x^2)*(1-2*x)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).

Original entry on oeis.org

1, 4, 15, 56, 209, 779, 2898, 10760, 39882, 147612, 545721, 2015721, 7440251, 27448008, 101217076, 373128151, 1375167467, 5067236766, 18669118971, 68774597447, 253334948791, 933111590565, 3436767111783, 12657552835178, 46616152422035, 171677402416052
Offset: 0

Views

Author

Philippe Deléham, Apr 10 2013

Keywords

Comments

A diagonal of the square array A223968.

Links

Crossrefs

Cf. A223968

Formula

a(n) = A223968(n,n+2).
G.f.: (1-3*x+x^2)*(1-2*x)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).
a(n) = 9*a(n-1) - 28*a(n-2) + 35*a(n-3) - 15*a(n-4) + a(n-5) with a(0) = 1, a(1) = 4, a(2) = 15, a(3) = 56, a(4) = 209.

A221862 Expansion of (1-3*x+x^2)*(1-x)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).

Original entry on oeis.org

1, 5, 21, 83, 319, 1208, 4535, 16932, 62986, 233702, 865513, 3201026, 11826582, 43660921, 161090910, 594092895, 2190225106, 8072519511, 29746921227, 109599320930, 403758993204, 1487294628182, 5478244777582, 20177275278559, 74313150143975, 273687550281967
Offset: 0

Views

Author

Philippe Deléham, Apr 10 2013

Keywords

Comments

A diagonal of the square array A223968.

Links

Crossrefs

Cf. A223968

Formula

a(n) = A223968(n,n+3).
G.f.: (1-3*x+x^2)*(1-x)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).
a(n) = 9*a(n-1) - 28*a(n-2) + 35*a(n-3) - 15*a(n-4) + a(n-5) with a(0) = 1, a(1) = 5, a(2) = 21, a(3) = 83, a(4) = 319.

A224509 Expansion of (1-x)*(1-2*x)*(1-3*x)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).

Original entry on oeis.org

1, 3, 10, 35, 125, 451, 1637, 5965, 21794, 79781, 292468, 1073296, 3941950, 14486721, 53264010, 195909180, 720769621, 2652351034, 9761957789, 35933354194, 132282020709, 487008295675, 1793068462212, 6602016250702, 24309222706566, 89511103102442
Offset: 0

Views

Author

Philippe Deléham, Apr 08 2013

Keywords

Comments

A diagonal of the square array A223968.

Links

Crossrefs

Cf. A223968

Formula

a(n) = A223968(n+1,n).
G.f.: (1-x)*(1-2*x)*(1-3*x)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).
a(n) = 9*a(n-1) - 28*a(n-2) + 35*a(n-3) - 15*a(n-4) + a(n-5) with a(0) = 1, a(1) = 3, a(2) = 10, a(3) = 35, a(4) = 125.
Showing 1-9 of 9 results.

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