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 1-4 of 4 results.

A168517 Expansion of g.f. (1/2)*( a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi(x, -n-1, 1) + c*2^(n+1)*(1-x)^(n+1)*LerchPhi(x, -n, 1/2) ), where a = -1, b = 1, and c = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 27, 27, 1, 1, 87, 260, 87, 1, 1, 263, 1828, 1828, 263, 1, 1, 779, 11131, 24746, 11131, 779, 1, 1, 2299, 62793, 267515, 267515, 62793, 2299, 1, 1, 6799, 338902, 2529377, 4753074, 2529377, 338902, 6799, 1, 1, 20175, 1780242, 21935526, 70068408, 70068408, 21935526, 1780242, 20175, 1
Offset: 0

Views

Author

Roger L. Bagula, Nov 28 2009

Keywords

Examples

			Triangle begins as:
  1;
  1,     1;
  1,     7,       1;
  1,    27,      27,        1;
  1,    87,     260,       87,        1;
  1,   263,    1828,     1828,      263,        1;
  1,   779,   11131,    24746,    11131,      779,        1;
  1,  2299,   62793,   267515,   267515,    62793,     2299,       1;
  1,  6799,  338902,  2529377,  4753074,  2529377,   338902,    6799,     1;
  1, 20175, 1780242, 21935526, 70068408, 70068408, 21935526, 1780242, 20175, 1;

Links

Crossrefs

Cf. A168518, A168549, A168551, A168552.

Programs

  • Mathematica
    p[x_, n_, a_, b_, c_]= (1/2)*(a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi[x,-n-1,1] + c*2^(n+1)*(1-x)^(n+1)*LerchPhi[x,-n,1/2]);
Table[CoefficientList[p[x,n,-1,1,1], x], {n,0,10}]//Flatten (* modified by G. C. Greubel, Mar 31 2022 *)
  • Sage
    def A168517(n,k,a,b,c): return (1/2)*( a*binomial(n,k) + sum( (-1)^(k-j)*(b*binomial(n+2, k-j)*(j+1)^(n+1) + 2*c*binomial(n+1,k-j)*(2*j+1)^n) for j in (0..k)) )
flatten([[A168517(n,k,-1,1,1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 31 2022

Formula

G.f.: (1/2)*( a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi(x, -n-1, 1) + c*2^(n+1)*(1 - x)^(n+1)*LerchPhi(x, -n, 1/2) ), where a = -1, b = 1, and c = 1.
From G. C. Greubel, Mar 31 2022: (Start)
T(n, k) = (1/2)*( a*binomial(n,k) + sum( (-1)^(k-j)*(b*binomial(n+2, k-j)*(j+1)^(n+1) + 2*c*binomial(n+1,k-j)*(2*j+1)^n) for j in (0..k)) ), with a = -1, b = 1, and c = 1.
T(n, n-k) = T(n, k). (End)

Extensions

Edited by G. C. Greubel, Mar 31 2022

A168549 Expansion of g.f. (1/2)*( a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi(x, -n-1, 1) + c*2^(n+1)*(1-x)^(n+1)*LerchPhi(x, -n, 1/2) ), where a = 31, b = -59, and c = 15, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 67, 67, 1, 1, 435, 1596, 435, 1, 1, 1951, 16476, 16476, 1951, 1, 1, 7383, 123243, 282258, 123243, 7383, 1, 1, 25507, 783537, 3435627, 3435627, 783537, 25507, 1, 1, 83595, 4543678, 34677285, 65518690, 34677285, 4543678, 83595, 1, 1, 265351, 24934378, 312192718, 1002545920, 1002545920, 312192718, 24934378, 265351, 1
Offset: 0

Views

Author

Roger L. Bagula, Nov 29 2009

Keywords

Examples

			Triangle begins as:
  1;
  1,     1;
  1,     3,       1;
  1,    67,      67,        1;
  1,   435,    1596,      435,        1;
  1,  1951,   16476,    16476,     1951,        1;
  1,  7383,  123243,   282258,   123243,     7383,       1;
  1, 25507,  783537,  3435627,  3435627,   783537,   25507,     1;
  1, 83595, 4543678, 34677285, 65518690, 34677285, 4543678, 83595, 1;

Links

Crossrefs

Cf. A001263. A168517, A168518, A168551, A168552.

Programs

  • Mathematica
    p[x_, n_, a_, b_, c_]= (1/2)*(a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi[x,-n-1,1] + c*2^(n+1)*(1-x)^(n+1)*LerchPhi[x,-n,1/2]);
Table[CoefficientList[p[x,n,31,-59,15], x], {n,0,10}]//Flatten (* modified by G. C. Greubel, Mar 31 2022 *)
  • Sage
    def A168549(n,k,a,b,c): return (1/2)*( a*binomial(n,k) + sum( (-1)^(k-j)*(b*binomial(n+2, k-j)*(j+1)^(n+1) + 2*c*binomial(n+1,k-j)*(2*j+1)^n) for j in (0..k)) )
flatten([[A168549(n,k,31,-59,15) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 31 2022

Formula

G.f.: (1/2)*( a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi(x, -n-1, 1) + c*2^(n+1)*(1 - x)^(n+1)*LerchPhi(x, -n, 1/2) ), where a = 31, b = -59, and c = 15.
From G. C. Greubel, Mar 31 2022: (Start)
T(n, k) = (1/2)*( a*binomial(n,k) + sum( (-1)^(k-j)*(b*binomial(n+2, k-j)*(j+1)^(n+1) + 2*c*binomial(n+1,k-j)*(2*j+1)^n) for j in (0..k)) ), with a = 31, b = -59, and c = 15.
T(n, n-k) = T(n, k). (End)

Extensions

Edited by G. C. Greubel, Mar 31 2022

A168551 Expansion of g.f. (1/2)*( a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi(x, -n-1, 1) + c*2^(n+1)*(1-x)^(n+1)*LerchPhi(x, -n, 1/2) ), where a = 1, b = -1, and c = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 19, 19, 1, 1, 65, 200, 65, 1, 1, 211, 1536, 1536, 211, 1, 1, 665, 9955, 22350, 9955, 665, 1, 1, 2059, 58521, 251931, 251931, 58521, 2059, 1, 1, 6305, 324322, 2441199, 4596954, 2441199, 324322, 6305, 1, 1, 19171, 1732438, 21480418, 68758180, 68758180, 21480418, 1732438, 19171, 1
Offset: 0

Views

Author

Roger L. Bagula, Nov 29 2009

Keywords

Examples

			Triangle begins as:
  1;
  1,     1;
  1,     5,       1;
  1,    19,      19,        1;
  1,    65,     200,       65,        1;
  1,   211,    1536,     1536,      211,        1;
  1,   665,    9955,    22350,     9955,      665,        1;
  1,  2059,   58521,   251931,   251931,    58521,     2059,       1;
  1,  6305,  324322,  2441199,  4596954,  2441199,   324322,    6305,     1;
  1, 19171, 1732438, 21480418, 68758180, 68758180, 21480418, 1732438, 19171, 1;

Links

Crossrefs

Cf. A001263, A132787.
Cf. A168517, A168518, A168549, A169552.

Programs

  • Mathematica
    p[x_, n_, a_, b_, c_]= (1/2)*(a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi[x,-n-1,1] + c*2^(n+1)*(1-x)^(n+1)*LerchPhi[x,-n,1/2]);
Table[CoefficientList[p[x,n,1,-1,1], x], {n,0,10}]//Flatten (* modified by G. C. Greubel, Mar 31 2022 *)
  • Sage
    def A168552(n,k,a,b,c): return (1/2)*( a*binomial(n,k) + sum( (-1)^(k-j)*(b*binomial(n+2, k-j)*(j+1)^(n+1) + 2*c*binomial(n+1,k-j)*(2*j+1)^n) for j in (0..k)) )
flatten([[A168552(n,k,1,-1,1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 31 2022

Formula

G.f.: (1/2)*( a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi(x, -n-1, 1) + c*2^(n+1)*(1 - x)^(n+1)*LerchPhi(x, -n, 1/2) ), where a = 1, b = -1, and c = 1.
From G. C. Greubel, Mar 31 2022: (Start)
T(n, k) = (1/2)*( a*binomial(n,k) + sum( (-1)^(k-j)*(b*binomial(n+2, k-j)*(j+1)^(n+1) + 2*c*binomial(n+1,k-j)*(2*j+1)^n) for j in (0..k)) ), with a = 1, b = -1, and c = 1.
T(n, n-k) = T(n, k). (End)

Extensions

Edited by G. C. Greubel, Mar 31 2022

A168552 Expansion of g.f. (1/2)*( a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi(x, -n-1, 1) + c*2^(n+1)*(1-x)^(n+1)*LerchPhi(x, -n, 1/2) ), where a = 3, b = -3, and c = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 11, 11, 1, 1, 43, 140, 43, 1, 1, 159, 1244, 1244, 159, 1, 1, 551, 8779, 19954, 8779, 551, 1, 1, 1819, 54249, 236347, 236347, 54249, 1819, 1, 1, 5811, 309742, 2353021, 4440834, 2353021, 309742, 5811, 1, 1, 18167, 1684634, 21025310, 67447952, 67447952, 21025310, 1684634, 18167, 1
Offset: 0

Views

Author

Roger L. Bagula, Nov 29 2009

Keywords

Examples

			Triangle begins as:
  1;
  1,     1;
  1,     3,       1;
  1,    11,      11,        1;
  1,    43,     140,       43,        1;
  1,   159,    1244,     1244,      159,        1;
  1,   551,    8779,    19954,     8779,      551,        1;
  1,  1819,   54249,   236347,   236347,    54249,     1819,       1;
  1,  5811,  309742,  2353021,  4440834,  2353021,   309742,    5811,     1;
  1, 18167, 1684634, 21025310, 67447952, 67447952, 21025310, 1684634, 18167, 1;

Links

Crossrefs

Cf. A001263, A168517, A168518, A168549, A168551.

Programs

  • Mathematica
    p[x_, n_, a_, b_, c_]= (1/2)*(a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi[x,-n-1,1] + c*2^(n+1)*(1-x)^(n+1)*LerchPhi[x,-n,1/2]);
Table[CoefficientList[p[x,n,3,-3,1], x], {n,0,10}]//Flatten (* modified by G. C. Greubel, Mar 31 2022 *)
  • Sage
    def A168552(n,k,a,b,c): return (1/2)*( a*binomial(n,k) + sum( (-1)^(k-j)*(b*binomial(n+2, k-j)*(j+1)^(n+1) + 2*c*binomial(n+1,k-j)*(2*j+1)^n) for j in (0..k)) )
flatten([[A168552(n,k,3,-3,1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 31 2022

Formula

G.f.: (1/2)*( a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi(x, -n-1, 1) + c*2^(n+1)*(1 - x)^(n+1)*LerchPhi(x, -n, 1/2) ), where a = 3, b = -3, and c = 1.
From G. C. Greubel, Mar 31 2022: (Start)
T(n, k) = (1/2)*( a*binomial(n,k) + sum( (-1)^(k-j)*(b*binomial(n+2, k-j)*(j+1)^(n+1) + 2*c*binomial(n+1,k-j)*(2*j+1)^n) for j in (0..k)) ), with a = 3, b = -3, and c = 1.
T(n, n-k) = T(n, k). (End)

Extensions

Edited by G. C. Greubel, Mar 31 2022
Showing 1-4 of 4 results.

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