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.

A060086 Convolution triangle A059594 with extra first column.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 5, 3, 1, 0, 3, 8, 9, 4, 1, 0, 3, 14, 19, 14, 5, 1, 0, 4, 20, 39, 36, 20, 6, 1, 0, 4, 30, 69, 85, 60, 27, 7, 1, 0, 5, 40, 119, 176, 160, 92, 35, 8, 1, 0, 5, 55, 189, 344, 376, 273, 133, 44, 9
Offset: 0

Views

Author

Wolfdieter Lang, Apr 06 2001

Keywords

Comments

Riordan array (1, x/((1+x)*(1-x)^2)). - Philippe Deléham, Feb 24 2012
Triangle, read by rows, given by (0, 1, 1, -2, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 24 2012

Examples

			{1}; {0,1}; {0,1,1}; {0,2,2,1}; ...
Triangle begins :
1
0, 1
0, 1, 1
0, 2, 2, 1
0, 2, 5, 3, 1
0, 3, 8, 9, 4, 1
0, 3, 14, 19, 14, 5, 1

Crossrefs

Cf. A059594,

Programs

  • Mathematica
    t[0, 0] = 1; t[, 0] = 0; t[n, m_] := Sum[ Sum[ Binomial[j, 2*j-3*k-m+n]*(-1)^(j-k)*Binomial[k, j], {j, 0, k}]*Binomial[m+k-1, m-1], {k, 0, n-m}]; Table[t[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jun 21 2013 *)

Formula

G.f.for column m >= 0: (x/((1-x^2)*(1-x)))^m.
T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k) - T(n-3,k) with T(n,0) = 0^n. - Philippe Deléham, Feb 24 2012
G.f.: (1-x-x^2+x^3)/(1-x-x^2+x^3-y*x). - Philippe Deléham, Feb 24 2012
Sum_{k, 0<=k<=n} T(n,k)*2^k = A181301(n). - Philippe Deléham, Feb 24 2012

A059595 Seventh column (m=6) of convolution triangle A059594(n,m).

Original entry on oeis.org

1, 7, 35, 133, 434, 1246, 3262, 7890, 17913, 38479, 78883, 155141, 294280, 540344, 963832, 1674568, 2841006, 4715970, 7673834, 12259142, 19254676, 29768972, 45355660, 68164628, 101143574, 148289946
Offset: 0

Views

Author

Wolfdieter Lang, Feb 02 2001

Keywords

Programs

  • Mathematica
    CoefficientList[Series[1/((1-x^2)(1-x))^7,{x,0,30}],x] (* Harvey P. Dale, Dec 04 2018 *)

Formula

G.f.: 1/((1-x^2)*(1-x))^7.
a(2*k)= binomial(k+10, 10)* (16*k^3+212*k^2+708*k+429)/(13*3*11);
a(2*k+1)= binomial(k+10, 10)*(16*k^3+316*k^2+1852*k+3003)/(13*3*11), k >= 0.

A059596 Eighth column (m=7) of convolution triangle A059594(n,m).

Original entry on oeis.org

1, 8, 44, 184, 654, 2040, 5772, 15048, 36693, 84448, 184976, 387872, 782680, 1525920, 2884560, 5302368, 9502014, 16635792, 28509272, 47902192, 79030348, 128192240, 204676056, 322002576, 499629966
Offset: 0

Views

Author

Wolfdieter Lang, Feb 02 2001

Keywords

Formula

G.f.: 1/((1-x^2)*(1-x))^8.
a(2*k)=binomial(k+11, 11)* (16*k^4+384*k^3+2936*k^2+7584*k+4095)/(15*7*13*3);
a(2*k+1)= binomial(k+12, 12)*8*(2*k+5)*(2*k+13)*(2*k+21)/(15*7*13), k >= 0.

A059597 Ninth column (m=8) of convolution triangle A059594(n,m).

Original entry on oeis.org

1, 9, 54, 246, 945, 3177, 9648, 26928, 70092, 171820, 399960, 889560, 1900380, 3915900, 7811280, 15129168, 28526562, 52480242, 94386908, 166242780, 287179794, 487227906, 812840976, 1334891664, 2160134700
Offset: 0

Views

Author

Wolfdieter Lang, Feb 02 2001

Keywords

Formula

G.f.: 1/((1-x^2)*(1-x))^9.
a(2*k)= binomial(k+13, 13)*(8*k^4+190*k^3+1414*k^2+3488*k+1785)/(17*15*7);
a(2*k+1)= binomial(k+13, 13)*(8*k^4+258*k^3+2842*k^2+12192*k+16065)/(17*15*7), k >= 0.

A059598 Tenth column (m=9) of convolution triangle A059594(n,m).

Original entry on oeis.org

1, 10, 65, 320, 1320, 4752, 15400, 45760, 126500, 328680, 809380, 1901120, 4282200, 9289840, 19482200, 39619008, 78337930, 150954980, 284060810, 522920640, 943206264, 1669294000, 2902420600, 4963400000
Offset: 0

Views

Author

Wolfdieter Lang, Feb 02 2001

Keywords

Links

Programs

  • Mathematica
    CoefficientList[Series[1/((1-x^2)(1-x))^10,{x,0,30}],x] (* or *) LinearRecurrence[{10,-35,20,195,-498,-15,1800,-2205,-2150,7001,-2260,-9785,10830,4845,-15504,4845,10830,-9785,-2260,7001,-2150,-2205,1800,-15,-498,195,20,-35,10,-1},{1,10,65,320,1320,4752,15400,45760,126500,328680,809380,1901120,4282200,9289840,19482200,39619008,78337930,150954980,284060810,522920640,943206264,1669294000,2902420600,4963400000,8356661300,13865072520,22688862900,36646948800,58465921800,92190872400},30] (* Harvey P. Dale, Oct 20 2021 *)

Formula

G.f.: 1/((1-x^2)*(1-x))^10.
a(2*k)= binomial(n+14, 14)*(2*n+15)*(8*n^4+240*n^3+2185*n^2+5775*n+2907)/(19*9*17*15);
a(2*k+1)= binomial(k+15, 15)*2*(8*k^4+256*k^3+2767*k^2+11504*k+14535)/(17*9*19), k >= 0

A059625 Eleventh column (m=10) of convolution triangle A059594.

Original entry on oeis.org

1, 11, 77, 407, 1793, 6875, 23661, 74503, 217789, 597311, 1549977, 3830619, 9065485, 20635967, 45353033, 96542523, 199597519, 401741989, 788857795, 1513922905, 2844244975, 5238604085, 9471346755
Offset: 0

Views

Author

Wolfdieter Lang, Feb 09 2001

Keywords

Formula

G.f.: 1/((1-x^2)*(1-x))^11.
a(2*k) = binomial(k + 16, 16)*(128*k^5 + 4768*k^4 + 62272*k^3 + 336488*k^2 + 673644*k + 305235)/(17*9*19*5*21); a(2*k + 1) = binomial(k + 16, 16)*(128*k^5 + 6112*k^4 + 107968*k^3 + 860312*k^2 + 2975580*k + 3357585)/(17*9*19*5*21), k >= 0.
Showing 1-6 of 6 results.

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