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.

A079513 Triangular array (a Riordan array) related to tennis ball problem, read by rows.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 3, 2, 1, 6, 6, 6, 3, 1, 0, 22, 16, 10, 4, 1, 53, 53, 53, 31, 15, 5, 1, 0, 211, 158, 105, 52, 21, 6, 1, 554, 554, 554, 343, 185, 80, 28, 7, 1, 0, 2306, 1752, 1198, 644, 301, 116, 36, 8, 1, 6362, 6362, 6362, 4056, 2304, 1106, 462, 161, 45, 9, 1
Offset: 0

Views

Author

N. J. A. Sloane, Jan 22 2003

Keywords

Comments

Riordan array (2/(2-x*c(x)+x*c(-x)), x*c(x)), with c(x) the g.f. of Catalan numbers (A000108). - Ralf Stephan, Dec 29 2013

Examples

			Triangle starts
     1;
     0,    1;
     1,    1,    1;
     0,    3,    2,    1;
     6,    6,    6,    3,    1;
     0,   22,   16,   10,    4,    1;
    53,   53,   53,   31,   15,    5,   1;
     0,  211,  158,  105,   52,   21,   6,   1;
   554,  554,  554,  343,  185,   80,  28,   7,  1;
     0, 2306, 1752, 1198,  644,  301, 116,  36,  8, 1;
  6362, 6362, 6362, 4056, 2304, 1106, 462, 161, 45, 9, 1;

Links

Crossrefs

First column is A066357 interspersed with 0's, 2nd column gives A079514.
Cf. A079514, A079515, A079516, A079517, A079518, A079519, A079520, A079521.

Programs

  • Mathematica
    c[t_]:= (1-Sqrt[1-4*t])/(2*t); d[t_]:= (1-(1+2*t)*Sqrt[1-4*t] -(1-2*t)*Sqrt[1+4*t] +Sqrt[1-16*t^2])/(4*t^2); g[t_, r_]:= d[t]*(t*c[t])^r; Table[SeriesCoefficient[Series[g[t, k], {t, 0, n}], n], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Jan 16 2019 *)

Extensions

Edited and more terms added by Ralf Stephan, Dec 29 2013

A079515 Coefficients related to tennis ball problem.

Original entry on oeis.org

1, 10, 105, 1198, 14506, 183284, 2390121, 31933830, 434920398, 6016012236, 84289034154, 1193717733900, 17060985356980, 245768668712296, 3564709196133737, 52015567131639798, 763050542202081318, 11246882679872658140, 166478073780305341390, 2473696423451621878180
Offset: 0

Views

Author

N. J. A. Sloane, Jan 22 2003

Keywords

Links

Crossrefs

Cf. A079513, A079514, A079516, A079517, A079518, A079519.

Programs

  • Mathematica
    c[t_]:= (1-Sqrt[1-4*t])/(2*t); d[t_]:= (1-(1+2*t)*Sqrt[1-4*t] -(1-2*t)*Sqrt[1+4*t] +Sqrt[1-16*t^2])/(4*t^2); g[t_, r_]:= d[t]*t^(r + 1)*c[t]^(r+3); CoefficientList[Series[g[t, 0], {t, 0, 60}], t][[2 ;; ;; 2]] (* G. C. Greubel, Jan 16 2019 *)

Formula

With c(t) = (1 - sqrt(1-4*t))/(2*t), d(t) = (1 -(1+2*t)*sqrt(1-4*t) -(1 - 2*t)*sqrt(1+4*t) + sqrt(1-16*t^2))/(4*t^2), and g(t, r) = d(t)*t^(r + 1)*c(t)^(r + 3) then the g.f. is given by the odd terms in the expansion of g(t,0) = t + 10*t^3 + 105*t^5 + 1198*t^7 + ... - G. C. Greubel, Jan 16 2019

Extensions

Terms a(5) onward added by G. C. Greubel, Jan 16 2019

A079516 Coefficients related to tennis ball problem.

Original entry on oeis.org

1, 15, 185, 2304, 29482, 386945, 5188169, 70803164, 980545070, 13747777966, 194776025482, 2784380900560, 40113386761524, 581823363803941, 8489505340500521, 124528817146723876, 1835299404114540102, 27163404479642455346, 403573421012802035630
Offset: 0

Views

Author

N. J. A. Sloane, Jan 22 2003

Keywords

Links

Crossrefs

Cf. A079513, A079514, A079515, A079517, A079518, A079519.

Programs

  • Mathematica
    c[t_]:= (1-Sqrt[1-4*t])/(2*t); d[t_]:= (1-(1+2*t)*Sqrt[1-4*t] - (1-2*t)*Sqrt[1+4*t] + Sqrt[1-16*t^2])/(4*t^2); g[t_, r_]:= d[t]*t^(r+1)*c[t]^(r+3); Drop[CoefficientList[Series[g[t, 1], {t, 0, 60}], t][[1 ;; ;; 2]], 1] (* G. C. Greubel, Jan 16 2019 *)

Formula

With c(t) = (1 - sqrt(1-4*t))/(2*t), d(t) = (1 -(1+2*t)*sqrt(1-4*t) -(1 - 2*t)*sqrt(1+4*t) + sqrt(1-16*t^2))/(4*t^2), and g(t, r) = d(t)*t^(r + 1)*c(t)^(r + 3) then the g.f. is given by the even terms in the expansion of g(t,1) = 1*t^2 + 15*t^4 + 185*t^6 + 2304*t^8 + ... - G. C. Greubel, Jan 16 2019

Extensions

Terms a(5) onward added by G. C. Greubel, Jan 16 2019

A079517 Coefficients related to tennis ball problem.

Original entry on oeis.org

1, 21, 301, 4088, 55354, 756059, 10442117, 145803900, 2056351566, 29262470042, 419730456306, 6062949606496, 88127311401876, 1288120149337735, 18922077118169717, 279209456350438708, 4136682188907493702, 61513664658938124486, 917795824360157700870
Offset: 0

Views

Author

N. J. A. Sloane, Jan 22 2003

Keywords

Links

Crossrefs

Cf. A079513, A079514, A079515, A079516, A079518.

Programs

  • Mathematica
    c[t_]:= (1-Sqrt[1-4*t])/(2*t); d[t_]:= (1-(1+2*t)*Sqrt[1-4*t] -(1-2*t)*Sqrt[1+4*t] +Sqrt[1-16*t^2])/(4*t^2); g[t_, r_]:= d[t]*t^(r+1)*c[t]^(r+3); Drop[CoefficientList[Series[g[t, 2], {t, 0, 60}], t][[2 ;; ;; 2]], 1] (* G. C. Greubel, Jan 16 2019 *)

Formula

With c(t) = (1 - sqrt(1-4*t))/(2*t), d(t) = (1 -(1+2*t)*sqrt(1-4*t) -(1 - 2*t)*sqrt(1+4*t) + sqrt(1-16*t^2))/(4*t^2), and g(t, r) = d(t)*t^(r + 1)*c(t)^(r + 3) then the g.f. is given by the odd terms in the expansion of g(t,2) = 1*t^3 + 21*t^5 + 301*t^7 + 4088*t^9 + ... - G. C. Greubel, Jan 16 2019

Extensions

Terms a(5) onward added by G. C. Greubel, Jan 16 2019

A079518 Coefficients related to tennis ball problem.

Original entry on oeis.org

1, 28, 462, 6832, 97957, 1394180, 19862674, 284156608, 4086496362, 59089988216, 858975619676, 12549322976672, 184195104642157, 2715174884250004, 40181870424263146, 596810833742837536, 8893877150513222014, 132947157383427373320, 1992954280253792526660
Offset: 0

Views

Author

N. J. A. Sloane, Jan 22 2003

Keywords

Links

Crossrefs

Cf. A079513, A079514, A079515, A079516, A079517, A079519.

Programs

  • Mathematica
    c[t_]:= (1-Sqrt[1-4*t])/(2*t); d[t_]:= (1-(1+2*t)*Sqrt[1-4*t] -(1- 2*t)*Sqrt[1+4*t] + Sqrt[1-16*t^2])/(4*t^2); g[t_, r_]:= d[t]*t^(r+1)*c[t]^(r+3); Drop[CoefficientList[Series[g[t, 3], {t,0,60}], t][[1;; ;;2]], 2] (* G. C. Greubel, Jan 16 2019 *)

Formula

With c(t) = (1 - sqrt(1-4*t))/(2*t), d(t) = (1 -(1+2*t)*sqrt(1-4*t) -(1 - 2*t)*sqrt(1+4*t) + sqrt(1-16*t^2))/(4*t^2), and g(t, r) = d(t)*t^(r + 1)*c(t)^(r + 3) then the g.f. is given by the even terms in the expansion of g(t,3) = 1*t^4 + 28*t^6 + 462*t^8 + 6832*t^10 + ... - G. C. Greubel, Jan 16 2019

Extensions

Terms a(5) onward added by G. C. Greubel, Jan 16 2019

A079519 Related to tennis ball problem.

Original entry on oeis.org

12, 284, 5436, 96768, 1664184, 28069444, 467722524, 7730252080, 127023181352, 2078332922360, 33894711502744, 551368536346176, 8950922822411504, 145068948446193428, 2347940754318431196, 37957946888159573968, 613052225104703442120, 9893099103451554441736
Offset: 1

Views

Author

N. J. A. Sloane, Jan 22 2003

Keywords

Examples

			G.f. = 12*t^2 + 284*t^4 + 5436*t^6 + 96768*t^8 + ... - _G. C. Greubel_, Jan 17 2019

Links

Crossrefs

Cf. A079513, A079514, A079515, A079516, A079517, A079518, A079520.

Programs

  • Mathematica
    f[t_]:= Sqrt[1-4*t]; g[t_]:= Sqrt[1+4*t]; S1[t_]:= (1+f[t]-2*f[t]^2)*(1- f[t])^5/(t^3*(f[t]^2-f[t])^2*(2+f[t]+g[t])^2); S3[t_]:= 4*(1-f[t])^2*(1 -g[t])^2*(f[t]^2-(1+2*t)*f[t]-(1-6*t)*g[t]+f[t]*g[t])/(t^3*(2+f[t]+ g[t])^2*(g[t]^2-f[t]-g[t]+f[t]*g[t])^2); W[t_]:= (S1[t]+S1[-t]+S3[t]+ S3[-t])/4; Drop[CoefficientList[Series[W[t], {t, 0, 50}], t][[1 ;; ;; 2]], 1] (* G. C. Greubel, Jan 17 2019 *)

Formula

Let f, g, S1 and S3 be given by f(t) = sqrt(1-4*t), g(t) = sqrt(1+4*t), S1(t) = (1+f(t)-2*f(t)^2)*(1- f(t))^5/(t^3*(f(t)^2-f(t))^2*(2+f(t)+g(t))^2), S3(t) = 4*(1-f(t))^2*(1 -g(t))^2*(f(t)^2-(1+2*t)*f(t)-(1-6*t)*g(t)+f(t)*g(t))/(t^3*(2+f(t)+ g(t))^2*(g(t)^2-f(t)-g(t)+ f(t)*g(t))^2). Now let W(t) be given by W(t) = (S1(t) + S1(-t) + S3(t) + S3(-t))/4. The g.f. is the expansion of W(t). - G. C. Greubel, Jan 17 2019

Extensions

Terms a(5) onward added by G. C. Greubel, Jan 17 2019
Showing 1-6 of 6 results.

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

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