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.

A089664 a(n) = S2(n,1), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.

Original entry on oeis.org

0, 4, 41, 306, 1966, 11540, 63726, 336700, 1720364, 8562024, 41718190, 199753004, 942561636, 4392660376, 20253510956, 92519626200, 419201709976, 1885719209936, 8428262686254, 37453751742604, 165575219275700, 728534225415864, 3191850894862564
Offset: 0

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Author

N. J. A. Sloane, Jan 04 2004

Keywords

Links

Crossrefs

Sequences of S2(n, t): A003583 (t=0), this sequence (t=1), A089665 (t=2), A089666 (t=3), A089667 (t=4), A089668 (t=5).
Cf. A000108, A089658, A089669.

Programs

  • Mathematica
    Table[(n*(3*n+5)*4^n -2*n*(n-1)*Binomial[2*n,n])/8, {n,0,40}] (* G. C. Greubel, May 25 2022 *)
  • PARI
    a(n)=n*(3*n+5)*2^(2*n-3) - 3*binomial(n+1,3)*binomial(2*n,n)/(n+1)/2 \\ Charles R Greathouse IV, Oct 23 2023
  • SageMath
    [(1/2)*(n*(3*n+5)*4^(n-1) -3*binomial(n+1, 3)*catalan_number(n)) for n in (0..40)] # G. C. Greubel, May 25 2022

Formula

a(n) = (1/8)*(n*(3*n+5)*4^n - 2*n*(n-1)*binomial(2*n, n)). (see Wang and Zhang, p. 338)
From G. C. Greubel, May 25 2022: (Start)
a(n) = (1/2)*( n*(3*n+5)*4^(n-1) - 3*binomial(n+1, 3)*Catalan(n) ).
G.f.: x*(4*(1-x) - 3*x*sqrt(1-4*x))/(1-4*x)^3.
E.g.f.: 2*x*(2 + 3*x)*exp(4*x) - (x^2/2)*(3*BesselI(0, 2*x) + 4*BesselI(1, 2*x) + BesselI(2, 2*x))*exp(2*x)). (End)

A089665 a(n) = S2(n,2), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.

Original entry on oeis.org

0, 4, 73, 788, 6630, 48120, 316526, 1940568, 11284380, 62968560, 339954670, 1786320184, 9176663028, 46248446608, 229285525420, 1120646918000, 5409322603896, 25824570392544, 122086747617198, 572130452101240, 2660063893120900, 12279619924999504, 56318986959592676
Offset: 0

Views

Author

N. J. A. Sloane, Jan 04 2004

Keywords

Links

Crossrefs

Sequences of S2(n, t): A003583 (t=0), A089664 (t=1), this sequence (t=2), A089666 (t=3), A089667 (t=4), A089668 (t=5).
Cf. A000108, A089658, A089669.

Programs

  • Maple
    S2:= (n, t) -> add(k^t*add(binomial(n, j), j = 0..k)^2, k = 0..n);
seq(S2(n, 2), n = 0..40);
  • Mathematica
    Table[(1/24)*(n*(n+1)*(7*n+5)*4^n -4*(n-1)*(3*n^2-2*n+1)*Binomial[2*n-2, n-1]), {n,0,40}] (* G. C. Greubel, May 25 2022 *)
  • SageMath
    [(n/6)*((n+1)*(7*n+5)*4^(n-1) -(n-1)*(3*n^2-2*n+1)*catalan_number(n-1)) for n in (0..40)] # G. C. Greubel, May 25 2022

Formula

a(n) = (1/24)*n*( (n+1)*(7*n+5)*4^n - 2*(n-1)*(3*n^2 - 2*n + 1)*binomial(2*n, n)/(2*n-1) ). (See Wang and Zhang, p. 338.)
From G. C. Greubel, May 25 2022: (Start)
a(n) = (n/6)*( (n+1)*(7*n+5)*4^(n-1) - (n-1)*(3*n^2 - 2*n + 1)*Catalan(n-1) ).
G.f.: x*(4*(1+3*x) - x*(3 + 2*x + 4*x^2)*sqrt(1-4*x))/(1-4*x)^4.
E.g.f.: x*(4 + 22*x + 56*x^2/3)*exp(4*x) + (x^2/6)*exp(2*x)*( -(9 + 62*x + 145*x^2 + 84*x^3)*f(x, 0) + (36 + 99*x - 32*x^2 - 84 x^3)*f(x, 1) + (45 + 270*x + 284*x^2 + 48*x^3)*f(x, 2) + x*(109 + 224*x + 78*x^2)*f(x, 3) + x^2*(53 + 36*x)*f(x, 4) + 6*x^3*f(x, 5) ), where f(x, n) = BesselI(n, 2*x). (End)

A089666 a(n) = S2(n,3), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.

Original entry on oeis.org

0, 4, 137, 2136, 23452, 209840, 1640346, 11648224, 76976048, 481048128, 2874897670, 16564931504, 92584313112, 504313834336, 2687067833492, 14045889333120, 72202366588096, 365713117287680, 1828223537042142, 9032706189007888, 44158716127799240, 213826835772518304
Offset: 0

Views

Author

N. J. A. Sloane, Jan 04 2004

Keywords

Links

Crossrefs

Sequences of S2(n, t): A003583 (t=0), A089664 (t=1), A089665 (t=2), this sequence (t=3), A089667 (t=4), A089668 (t=5).
Cf. A000108, A089658, A089669.

Programs

  • Maple
    S2:= (n, t) -> add(k^t*add(binomial(n, j), j = 0..k)^2, k = 0..n);
seq(S2(n, 3), n = 0..40);
  • Mathematica
    Table[n*(15*n^3+30*n^2+21*n-2)*4^(n-3) -(n-1)^2*n^2*(n+1)*Binomial[2*n,n]/(8*(2*n -1)), {n, 0, 40}] (* G. C. Greubel, May 25 2022 *)
  • SageMath
    [n*(15*n^3+30*n^2+21*n-2)*4^(n-3) - 9*binomial(n+1, 3)^2 * catalan_number(n-1)/(n+1) for n in (0..40)] # G. C. Greubel, May 25 2022

Formula

a(n) = n*(15*n^3 + 30*n^2 + 21*n - 2)*4^(n-3) - (n-1)^2*n^2*(n+1)*binomial(2*n, n)/(8*(2*n-1)). (See Wang and Zhang, p. 338.)
From G. C. Greubel, May 25 2022: (Start)
a(n) = n*(15*n^3 + 30*n^2 + 21*n - 2)*4^(n-3) - 9*binomial(n+1, 3)^2 * Catalan(n- 1)/(n+1).
G.f.: x*(4*(1 + 15*x + 12*x^2 + 8*x^3) - 3*x*(1 + 6*x - 6*x^2 + 4*x^3)*sqrt(1-4*x))/(1-4*x)^5. (End)

Extensions

Name changed by G. C. Greubel, May 25 2022

A089667 a(n) = S2(n,4), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.

Original entry on oeis.org

0, 4, 265, 5984, 85722, 944904, 8771462, 72095520, 541127988, 3785356752, 25032083230, 158102986624, 961123994220, 5656943319664, 32386277835772, 181019819948864, 990793669704552, 5323620638111136, 28137973407708174, 146552649537716992
Offset: 0

Views

Author

N. J. A. Sloane, Jan 04 2004

Keywords

Links

Crossrefs

Sequences of S2(n, t): A003583 (t=0), A089664 (t=1), A089665 (t=2), A089666 (t=3), this sequence (t=4), A089668 (t=5).
Cf. A000108, A089658, A089669.

Programs

  • Mathematica
    Table[(1/30)*(n*(n+1)*(93*n^3+132*n^2+53*n-38)*4^(n-2) -(n-1)*(15*n^5-99*n^3 + 116*n^2-34*n+6)*CatalanNumber[n-2]), {n,0,40}] (* G. C. Greubel, May 25 2022 *)
CoefficientList[Series[x*( 4*(1 + 43*x + 160*x^2 + 96*x^3) - x*(3 + 62*x - 72*x^2 + 96*x^3 - 224*x^4 + 144*x^5)*Sqrt[1-4*x] )/(1-4*x)^6, {x,0,35}], x] (* Georg Fischer, Nov 09 2022 *)
  • SageMath
    [(1/30)*(n*(n+1)*(93*n^3+132*n^2+53*n-38)*4^(n-2) - (n-1)*(15*n^5 - 99*n^3+116*n^2-34*n+6)*catalan_number(n-2) ) for n in (0..40)] # G. C. Greubel, May 25 2022

Formula

a(n) = (1/480)*( n*(n+1)*(93*n^3 + 132*n^2 + 53*n - 38)*4^n - 4*n*(n-1)*(15*n^5 - 99*n^3 + 116*n^2 - 34*n + 6)*binomial(2*n, n)/((2*n-1)*(2*n-3)) ). (See Wang and Zhang, p. 338.)
From G. C. Greubel, May 25 2022: (Start)
a(n) = (1/30)*( n*(n+1)*(93*n^3 + 132*n^2 + 53*n - 38)*4^(n-2) - (n-1)*(15*n^5 - 99*n^3 + 116*n^2 - 34*n + 6)*Catalan(n-2) ).
G.f.: x*( 4*(1 + 43*x + 160*x^2 + 96*x^3) - x*(3 + 62*x - 72*x^2 + 96*x^3 - 224*x^4 + 144*x^5)*sqrt(1-4*x) )/(1-4*x)^6. [Typo corrected by Georg Fischer, Nov 09 2022] (End)
Showing 1-4 of 4 results.

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

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