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.

A079522 Diagonal of triangular array in A079520.

Original entry on oeis.org

0, 1, 3, 10, 31, 105, 343, 1198, 4056, 14506, 50350, 183284, 647809, 2390121, 8564543, 31933830, 115664164, 434920398, 1588917802, 6016012236, 22134533070, 84289034154, 311957090678, 1193717733900, 4440128821376, 17060985356980, 63732279047612, 245768668712296, 921501110779045
Offset: 0

Views

Author

N. J. A. Sloane, Jan 22 2003

Keywords

Examples

			G.f. = 0 + 1*t + 3*t^2 + 10*t^3 + 31*t^4 + ... - _G. C. Greubel_, Jan 17 2019

Links

Crossrefs

Also diagonal of triangular array in A079521.
Cf. A079513, A079520.

Programs

  • Maple
    F := proc(t) (1-4*t^2-(1+2*t)*sqrt(1-4*t)-(1-2*t)*sqrt(1+4*t)+ sqrt(1-16*t^2))/4/t^3 ; end: d := proc(t) 1+t*F(t) ; end: C := proc(t) (1-sqrt(1-4*t))/2/t ; end: A079521 := proc(h,r) d(t)*t^(r+1)*(C(t))^(r+3) ; expand(%) ; coeftayl(%,t=0,h) ; end: A079522 := proc(n) A079521(n,0) ; end: for n from 0 do printf("%d\n",A079522(n)) ; od: # R. J. Mathar, Sep 20 2009
  • 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*(t*c[t]^3 +2*r*c[t]); CoefficientList[Series[g[t, 0], {t, 0, 50}], t] (* G. C. Greubel, Jan 17 2019 *)

Formula

Let c, d, and g be given by: 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*c(t))^r*(t*c(t)^3 + 2*r*c(t)) then the g.f. is given by the expansion of g(t,0). - G. C. Greubel, Jan 17 2019
a(n) ~ 2^(2*n + 1/2) * (9*sqrt(2) - 10 + (41*sqrt(2) - 58)*(-1)^n) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 31 2025

Extensions

More terms from R. J. Mathar, Sep 20 2009
Terms a(23) onward added by G. C. Greubel, Jan 17 2019

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

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

A079521 Triangular array related to tennis ball problem, read by rows.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 10, 16, 13, 6, 31, 47, 45, 25, 8, 105, 158, 145, 96, 41, 10, 343, 501, 500, 340, 175, 61, 12, 1198, 1752, 1673, 1226, 676, 288, 85, 14, 4056, 5808, 5898, 4326, 2569, 1205, 441, 113, 16, 14506, 20868, 20312, 15608, 9526, 4836, 1987, 640, 145, 18, 50350, 71218, 73000, 55696, 35448, 18800, 8418, 3090, 891, 181, 20
Offset: 0

Views

Author

N. J. A. Sloane, Jan 22 2003

Keywords

Examples

			0.
1,   2.
3,   5,   4.
10,  16,  13,  6.
31,  47,  45,  25, 8.
105, 158, 145, 96, 41, 10. ...

Links

Crossrefs

Leading diagonal gives A079522.
Cf. A079513, 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*(t*c[t]^3 +2*r*c[t]); Table[SeriesCoefficient[Series[g[t, k], {t, 0, n}], n], {n, 0, 10}, {k, 0, n}] (* G. C. Greubel, Jan 17 2019 *)

Formula

Let c, d, and g be given by: 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*c(t))^r*(t*c(t)^3 + 2*r*c(t)) then the rows are calculated by the expansion of g(t,k) for n>=0, 0 <= k <= n. - G. C. Greubel, Jan 17 2019

Extensions

Terms a(28) onward added by G. C. Greubel, Jan 17 2019
Showing 1-4 of 4 results.

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

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