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 1-9 of 9 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

Views

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)

A089663 a(n) = S1(n, 6), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).

Original entry on oeis.org

0, 2, 259, 6284, 77180, 646960, 4235864, 23313408, 112793088, 493969920, 1998346240, 7577934848, 27232132096, 93517705216, 308908943360, 986642513920, 3059995508736, 9247515082752, 27310549696512, 79012328898560, 224396746424320, 626707269681152
Offset: 0

Views

Author

N. J. A. Sloane, Jan 04 2004

Keywords

Links

Crossrefs

Sequences of S1(n, t): A001792 (t=0), A089658 (t=1), A089659 (t=2), A089660 (t=3), A089661 (t=4), A089662 (t=5), this sequence (t=6).
Cf. A089664, A089669.

Programs

  • Magma
    [2^(n-7)*n*(381*n^6+1302*n^5+1302*n^4+420*n^3-707*n^2-378*n+368)/21: n in [0..40]]; // G. C. Greubel, May 24 2022
  • Mathematica
    LinearRecurrence[{16,-112,448,-1120,1792,-1792,1024,-256}, {0, 2, 259, 6284, 77180, 646960, 4235864, 23313408}, 40] (* G. C. Greubel, Jun 22 2016 *)
  • SageMath
    [n*(n+1)*(381*n^5 +921*n^4 +381*n^3 +39*n^2 -746*n +368)*2^(n-7)/21 for n in (0..40)] # G. C. Greubel, May 24 2022

Formula

a(n) = (1/21)*n*(n+1)*(381*n^5 + 921*n^4 + 381*n^3 + 39*n^2 - 746*n + 368) * 2^(n-7). (See Wang and Zhang, p. 334.)
From Chai Wah Wu, Jun 21 2016: (Start)
a(n) = 16*a(n-1) - 112*a(n-2) + 448*a(n-3) - 1120*a(n-4) + 1792*a(n-5) - 1792*a(n-6) + 1024*a(n-7) - 256*a(n-8) for n > 7.
G.f.: x*(2 + 227*x + 2364*x^2 + 4748*x^3 + 2096*x^4 - 72*x^5)/(1 - 2*x)^8. (End)
a(n) = 2^(n-7)*n*(381*n^6 + 1302*n^5 + 1302*n^4 + 420*n^3 - 707*n^2 - 378*n + 368)/21. - Ilya Gutkovskiy, Jun 21 2016
E.g.f.: (1/42)*x*(84 + 5271*x + 33278*x^2 + 57855*x^3 + 37086*x^4 + 9303*x^5 +
762*x^6)*exp(2*x). - G. C. Greubel, May 24 2022

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)

A089668 a(n) = S2(n,5), 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, 521, 17136, 320716, 4356560, 48024786, 456843520, 3893995184, 30487086144, 223052123830, 1544098243424, 10208488021176, 64917814932256, 399310478637476, 2386386863086080, 13906802738650816, 79261768839946496, 442921922267640894
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), A089667 (t=4), this sequence (t=5).
Cf. A000108, A089658, A089669.

Programs

  • Mathematica
    Table[(1/2)*(n*(21*n^5+61*n^4+55*n^3+15*n^2-28*n+4)*4^(n-3) -(n-1)*(3*n-5)*(n^2 + 4*n-6)*Binomial[n+1, 3]*CatalanNumber[n-2]), {n, 0, 40}] (* G. C. Greubel, May 25 2022 *)
  • SageMath
    [(1/2)*(n*(21*n^5 + 61*n^4 + 55*n^3 + 15*n^2 - 28*n + 4)*4^(n-3) - (n-1)*(3*n-5)*(n^2 + 4*n - 6)*binomial(n+1, 3)*catalan_number(n-2)) for n in (0..40)] # G. C. Greubel, May 25 2022

Formula

a(n) = (1/128)*n*(21*n^5 + 61*n^4 + 55*n^3 + 15*n^2 - 28*n + 4)*4^n - (1/48)*n^2*(n-1)^2*(3*n-5)*(n^2 + 4*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) = (1/2)*(n*(21*n^5 + 61*n^4 + 55*n^3 + 15*n^2 - 28*n + 4)*4^(n-3) - (n-1)*(3*n-5)*(n^2 + 4*n - 6)*binomial(n+1, 3)*Catalan(n-2)).
G.f.: x*( 4*(1 + 103*x + 1012*x^2 + 1688*x^3 + 512*x^4 - 256*x^5) - 3*x*(1 + 54*x + 26*x^2 - 156*x^3 - 104*x^4 + 320*x^5 -240*x^6)*sqrt(1-4*x) )/(1-4*x)^7. (End)

A089670 a(n) = S3(n,2), where S3(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^3.

Original entry on oeis.org

0, 8, 283, 6044, 101360, 1470640, 19361174, 237684384, 2768042208, 30935313600, 334481353690, 3519672963752, 36206551801264, 365363625058432, 3626585989411280, 35485636769545600, 342894590805622656, 3276865150482420480, 31008279252965786178
Offset: 0

Views

Author

N. J. A. Sloane, Jan 04 2004

Keywords

Links

Crossrefs

Sequences of S3(n, t): A007403 (t=0), A089669 (t=1), this sequence (t=2), A089671 (t=3), A089672 (t=4).
Cf. A089658, A089664.

Programs

  • Mathematica
    a[n_]:= a[n]= Sum[k^2*(Sum[Binomial[n, j], {j,0,k}])^3, {k,0,n}];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, May 26 2022 *)
  • SageMath
    def A089670(n): return sum(k^2*(sum(binomial(n,j) for j in (0..k)))^3 for k in (0..n))
[A089670(n) for n in (0..40)] # G. C. Greubel, May 26 2022

Formula

a(n) = Sum_{k=0..n} k^2 *(Sum_{j=0..k} binomial(n,j))^3. - G. C. Greubel, May 26 2022
a(n) ~ 7/24 * 8^n * n^3 * (1 - 9/(14*sqrt(Pi*n)) + (12/7 - 3^(3/2)/(14*Pi))/n). - Vaclav Kotesovec, May 27 2022

A089671 a(n) = S3(n,3), where S3(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^3.

Original entry on oeis.org

0, 8, 539, 16632, 364042, 6510160, 101817234, 1447146512, 19144522160, 239513659776, 2865559784050, 33052451375152, 369790434398988, 4031805422883680, 42996629236138928, 449821323139340160, 4627609615665499456, 46907404618252667392, 469254490696475078130
Offset: 0

Views

Author

N. J. A. Sloane, Jan 04 2004

Keywords

Links

Crossrefs

Sequences of S3(n, t): A007403 (t=0), A089669 (t=1), A089670 (t=2), this sequence (t=3), A089672 (t=4).
Cf. A089658, A089664.

Programs

  • Mathematica
    a[n_]:= a[n]= Sum[k^3*(Sum[Binomial[n, j], {j,0,k}])^3, {k,0,n}];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, May 26 2022 *)
  • SageMath
    def A089671(n): return sum(k^3*(sum(binomial(n,j) for j in (0..k)))^3 for k in (0..n))
[A089671(n) for n in (0..40)] # G. C. Greubel, May 26 2022

Formula

a(n) = Sum_{k=0..n} k^3 *(Sum_{j=0..k} binomial(n,j))^3. - G. C. Greubel, May 26 2022
a(n) ~ 15 * 2^(3*n-6) * n^4 * (1 - 2/(5*sqrt(Pi*n)) + (2 - sqrt(3)/(5*Pi))/n). - Vaclav Kotesovec, May 27 2022

A089672 a(n) = S3(n,4), where S3(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^3.

Original entry on oeis.org

0, 8, 1051, 47024, 1343372, 29595904, 549599246, 9039987264, 135800368320, 1901346478080, 25165027679242, 318105020914208, 3870088369412824, 45584244411107584, 522235732874214800, 5840992473138691072, 63970901725419781632, 687749464543749095424, 7273214936974305201570
Offset: 0

Views

Author

N. J. A. Sloane, Jan 04 2004

Keywords

Links

Crossrefs

Sequences of S3(n, t): A007403 (t=0), A089669 (t=1), A089670 (t=2), A089671 (t=3), this sequence (t=4).
Cf. A089658, A089664.

Programs

  • Maple
    S3:= (n, t) -> add(k^t*add(binomial(n, j), j = 0..k)^3, k = 0..n);
seq(S3(n, 4), n = 0..40);
  • Mathematica
    a[n_]:= a[n]= Sum[k^4*(Sum[Binomial[n, j], {j,0,k}])^3, {k,0,n}];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, May 26 2022 *)
  • SageMath
    def A089672(n): return sum(k^4*(sum(binomial(n,j) for j in (0..k)))^3 for k in (0..n))
[A089672(n) for n in (0..40)] # G. C. Greubel, May 26 2022

Formula

a(n) = Sum_{k=0..n} k^4 *(Sum_{j=0..k} binomial(n,j))^3. - G. C. Greubel, May 26 2022
a(n) ~ 31 * 2^(3*n - 5) * n^5 / 5 * (1 - 15/(62*sqrt(Pi*n)) + (75 - 5*sqrt(3)/Pi) / (31*n)). - Vaclav Kotesovec, May 27 2022
Showing 1-9 of 9 results.

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

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