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 1-10 of 30 results. Next

A055831 T(n,n-4), where T is the array in A055830.

Original entry on oeis.org

5, 15, 31, 54, 85, 125, 175, 236, 309, 395, 495, 610, 741, 889, 1055, 1240, 1445, 1671, 1919, 2190, 2485, 2805, 3151, 3524, 3925, 4355, 4815, 5306, 5829, 6385, 6975, 7600, 8261, 8959, 9695, 10470, 11285, 12141, 13039
Offset: 4

Views

Author

Clark Kimberling, May 28 2000

Keywords

Links

Crossrefs

Cf. A055830.

Programs

  • GAP
    List([4..40], n-> (n-3)*(n^2+6*n-10)/6); # G. C. Greubel, Jan 21 2020
  • Magma
    [(n-3)*(n^2+6*n-10)/6: n in [4..40]]; // G. C. Greubel, Jan 21 2020
  • Maple
    seq( (n-3)*(n^2+6*n-10)/6, n=4..40); # G. C. Greubel, Jan 21 2020
  • Mathematica
    Table[(n-3)*(n^2+6*n-10)/6, {n,4,40}] (* G. C. Greubel, Jan 21 2020 *)
  • PARI
    a(n) = (n-3)*(n^2+6*n-10)/6; \\ G. C. Greubel, Jan 21 2020
  • Sage
    [(n-3)*(n^2+6*n-10)/6 for n in (4..40)] # G. C. Greubel, Jan 21 2020

Formula

a(n) = (n-3)*(n^2+6*n-10)/6, n>3.
G.f.: x^4*(5-5*x+x^2)/(1-x)^4. - R. J. Mathar, Mar 13 2016
E.g.f.: (-30 - 6*x + 3*x^2 + (30 - 24*x + 6*x^2 + x^3)*exp(x))/6. - G. C. Greubel, Jan 21 2020

A055834 a(n) = T(2n,n), where T is the array in A055830.

Original entry on oeis.org

1, 1, 4, 18, 85, 413, 2044, 10248, 51876, 264550, 1357070, 6994780, 36196706, 187938842, 978599560, 5108177816, 26721644973, 140050505085, 735254208670, 3865837887450, 20353393741065, 107290306033845, 566194674179160, 2990958274811520, 15814562990604300, 83690040760923168
Offset: 0

Views

Author

Clark Kimberling, May 28 2000

Keywords

Links

Crossrefs

Cf. A001002, A055830, A055835.

Programs

  • GAP
    List([0..30], n-> Sum([0..n], k-> Binomial(n+k-1,n)*Binomial(k,n-k)) ); # G. C. Greubel, Jan 21 2020
  • Magma
    [&+[(Binomial(n+k-1, n)*Binomial(k, n-k)): k in [0..n]]: n in [0..30]]; // Vincenzo Librandi, Sep 21 2015
  • Maple
    seq( add(binomial(n+k-1,n)*binomial(k,n-k), k=0..n), n=0..30); # G. C. Greubel, Jan 21 2020
  • Mathematica
    Table[Sum[Binomial[n+k-1,n]Binomial[k,n-k],{k,0,n}],{n,0,30}] (* Harvey P. Dale, Oct 03 2011 *)
  • Maxima
    b(n):= sum(binomial(n+k, k)*binomial(k, n-k), k,ceiling(n/2),n)/(n+1);
B(x):= sum(b(i)*x^(i),i,0,30);
makelist(coeff(taylor(x*diff(B(x),x)-x*diff(B(x),x)/B(x)+B(x), x,0,20), x,n), n,0,20); /* Vladimir Kruchinin, Sep 21 2015 */
  • PARI
    a(n) = sum(k=0,n,binomial(n+k-1,n)*binomial(k,n-k)); \\ Joerg Arndt, May 06 2013
  • Sage
    [sum(binomial(n+k-1,n)*binomial(k,n-k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Jan 21 2020

Formula

a(n) = Sum_{k=0..n} binomial(n+k-1,n)*binomial(k,n-k). - Max Alekseyev, Jun 17 2007
Recurrence: 5*(n-1)*n*a(n) = 2*(n-1)*(11*n-3)*a(n-1) + 3*(3*n-5)*(3*n-4)*a(n-2). - Vaclav Kotesovec, Nov 19 2012
a(n) ~ 27^n/5^n*sqrt(2/(15*Pi*n)). - Vaclav Kotesovec, Nov 19 2012
a(n) = A055835(n)/3 for n>=1. - Philippe Deléham, Jan 25 2014
G.f.: x*B'(x)-x*B'(x)/B(x)+B(x), where B(x) is g.f. of A001002. - Vladimir Kruchinin, Sep 20 2015

A055835 T(2n+1,n), where T is the array in A055830.

Original entry on oeis.org

1, 3, 12, 54, 255, 1239, 6132, 30744, 155628, 793650, 4071210, 20984340, 108590118, 563816526, 2935798680, 15324533448, 80164934919, 420151515255, 2205762626010, 11597513662350, 61060181223195, 321870918101535
Offset: 0

Views

Author

Clark Kimberling, May 28 2000

Keywords

Links

Crossrefs

Cf. A055830, A055834.

Programs

  • GAP
    Concatenation([1], List([1..30], n-> 3*Sum([0..n], k-> Binomial(n+k-1, n) *Binomial(k, n-k) ))); # G. C. Greubel, Jan 21 2020
  • Magma
    [1] cat [3*(&+[Binomial(n+k-1, n)*Binomial(k, n-k): k in [0..n]]): n in [1..30]]; // G. C. Greubel, Jan 21 2020
  • Maple
    seq( `if`(n=0, 1, 3*add(binomial(n+k-1, n)*binomial(k, n-k), k=0..n)), n=0..30); # G. C. Greubel, Jan 21 2020
  • Mathematica
    Table[If[n==0, 1, 3*Sum[Binomial[k, n-k]*Binomial[n+k-1, n], {k,0,n}]], {n,0,30}] (* G. C. Greubel, Jan 21 2020 *)
  • Maxima
    a(n):=sum((sum(binomial(i,k)*binomial(i+1,n-i),i,k,n))*binomial(n,k), k,0,n); /* Vladimir Kruchinin, Mar 01 2014 */
  • PARI
    a(n) = if(n==0, 1, 3*sum(k=0, n, binomial(n+k-1, n)*binomial(k, n-k)) ); \\ Joerg Arndt, Mar 01 2014
  • Sage
    [1]+[3*sum(binomial(n+k-1,n)*binomial(k,n-k) for k in (0..n)) for n in (1..30)] # G. C. Greubel, Jan 21 2020

Formula

a(n) = 3*A055834(n) for n>=1. - Philippe Deléham, Jan 25 2014
a(n) = Sum_{k=0..n} Sum_{i=k..n} binomial(i,k)*binomial(i+1,n-i)*binomial(n,k). - Vladimir Kruchinin, Mar 01 2014

A055832 T(n,n-5), where T is the array in A055830.

Original entry on oeis.org

8, 30, 73, 145, 255, 413, 630, 918, 1290, 1760, 2343, 3055, 3913, 4935, 6140, 7548, 9180, 11058, 13205, 15645, 18403, 21505, 24978, 28850, 33150, 37908, 43155, 48923, 55245, 62155, 69688, 77880, 86768, 96390
Offset: 5

Views

Author

Clark Kimberling, May 28 2000

Keywords

Links

Crossrefs

Cf. A055830.

Programs

  • GAP
    List([5..40], n-> (n-3)*(n-4)*(n^2+13*n+6)/24); # G. C. Greubel, Jan 21 2020
  • Magma
    [(n-3)*(n-4)*(n^2+13*n+6)/24: n in [5..40]]; // G. C. Greubel, Jan 21 2020
  • Maple
    seq( (n-3)*(n-4)*(n^2+13*n+6)/4!, n=5..40); # G. C. Greubel, Jan 21 2020
  • Mathematica
    Table[(n-3)*(n-4)*(n^2+13*n+6)/4!, {n,5,40}] (* G. C. Greubel, Jan 21 2020 *)
  • PARI
    a(n) = (n-3)*(n-4)*(n^2+13*n+6)/4!; \\ G. C. Greubel, Jan 21 2020
  • Sage
    [(n-3)*(n-4)*(n^2+13*n+6)/24 for n in (5..40)] # G. C. Greubel, Jan 21 2020

Formula

a(n) = (n-3)*(n-4)*(n^2+13*n+6)/24, n>4, sign corrected Mar 13 2016.
G.f.: x^5*(4-3*x)*(2-x)/(1-x)^5. - R. J. Mathar, Mar 13 2016
E.g.f.: (-72 -120*x -36*x^2 + (72 +48*x -48*x^2 +12*x^3 +x^4)*exp(x))/24. - G. C. Greubel, Jan 21 2020

A055833 T(n,n-6), where T is the array in A055830.

Original entry on oeis.org

13, 58, 162, 361, 701, 1239, 2044, 3198, 4797, 6952, 9790, 13455, 18109, 23933, 31128, 39916, 50541, 63270, 78394, 96229, 117117, 141427, 169556, 201930, 239005, 281268, 329238, 383467, 444541, 513081
Offset: 6

Views

Author

Clark Kimberling, May 28 2000

Keywords

Links

Crossrefs

Cf. A055830.

Programs

  • GAP
    List([6..40], n-> (n-5)*(n-4)*(n^3 +19*n^2 +6*n -156)/120 ); # G. C. Greubel, Jan 21 2020
  • Magma
    [(n-5)*(n-4)*(n^3 +19*n^2 +6*n -156)/120: n in [6..40]]; // G. C. Greubel, Jan 21 2020
  • Maple
    seq( (n-5)*(n-4)*(n^3 +19*n^2 +6*n -156)/120, n=6..40); # G. C. Greubel, Jan 21 2020
  • Mathematica
    Table[(n-5)*(n-4)*(n^3 +19*n^2 +6*n -156)/120, {n,6,40}] (* G. C. Greubel, Jan 21 2020 *)
  • PARI
    a(n) = (n-5)*(n-4)*(n^3 +19*n^2 +6*n -156)/120; \\ G. C. Greubel, Jan 21 2020
  • Sage
    [(n-5)*(n-4)*(n^3 +19*n^2 +6*n -156)/120 for n in (6..40)] # G. C. Greubel, Jan 21 2020

Formula

From R. J. Mathar, Mar 13 2016: (Start)
G.f.: x^6*(13 -20*x +9*x^2 -x^3)/(1-x)^6.
a(n) = (n-5)*(n-4)*(n^3 +19*n^2 +6*n -156)/120. (End)
E.g.f.: (3120 + 1560*x + 180*x^2 - 20*x^3 - (3120 - 1560*x + 180*x^2 + 60*x^3 - 20*x^4 - x^5)*exp(x))/120. - G. C. Greubel, Jan 21 2020

A055836 T(2n+2, n), where T is the array in A055830.

Original entry on oeis.org

2, 7, 31, 145, 701, 3458, 17298, 87417, 445225, 2281565, 11750245, 60763950, 315315014, 1641046720, 8562466432, 44775095601, 234594444741, 1231249999640, 6472043549400, 34067089542255, 179543120927115
Offset: 0

Views

Author

Clark Kimberling, May 28 2000

Keywords

Links

Programs

  • Magma
    [Binomial(2*n+1, n) + (&+[Binomial(j+1, n-j+1)*Binomial(n+j, n): j in [Ceiling(n/2)..n]]): n in [0..25]]; // G. C. Greubel, Jun 09 2019
  • Mathematica
    a[n_]:= Binomial[2n+1, n] + Sum[Binomial[i+1, n-i+1] Binomial[n+i, n], {i, Ceiling[n/2], n}];
Array[a, 21, 0]  (* Jean-François Alcover, Jun 03 2019, after Vladimir Kruchinin *)
  • Maxima
    a(n):=binomial(2*n+1,n)+sum(binomial(i+1,n-i+1)*binomial(n+i,n),i,ceiling((n)/2),n); /* Vladimir Kruchinin, Nov 26 2014 */
  • PARI
    {a(n) = binomial(2*n+1,n) + sum(j=ceil(n/2), n, binomial(j+1, n-j+1)*binomial(n+j,n))}; \\ G. C. Greubel, Jun 09 2019
  • Sage
    def A055836(n):
    c = ceil(n/2)
    b = binomial(c+1,n-c+1)*binomial(n+c,n)
    h = hypergeometric([1,c+2,-n+c-1,n+c+1],[c+1,-n/2+c+1/2,-n/2+c+1],-1/4)
    return b*h.simplify_hypergeometric()
[A055836(n) for n in range(21)] # Peter Luschny, Nov 28 2014

Formula

a(n) = binomial(2*n+1,n) + Sum_{i=ceiling(n/2)..n} binomial(i+1,n-i+1)*binomial(n+i,n). - Vladimir Kruchinin, Nov 26 2014
a(n) = C(c+1,n-c+1)*C(n+c,n)*hypergeom([1,c+2,-n+c-1,n+c+1],[c+1,-n/2+c+1/2,-n/2+c+1],-1/4) where c=ceiling(n/2). - Peter Luschny, Nov 28 2014
Conjecture: 5*n*(n+1)*(7*n-5)*a(n) - n*(154*n^2+2*n-77)*a(n-1) - 3*(3*n-4)*(7*n+2)*(3*n-2)*a(n-2) = 0. - R. J. Mathar, Mar 13 2016

A055837 T(2n+3,n), where T is the array in A055830.

Original entry on oeis.org

3, 15, 73, 361, 1806, 9122, 46425, 237721, 1223365, 6321965, 32784830, 170528190, 889291352, 4648068192, 24342384337, 127707864849, 671047979300, 3531026714720, 18603737992455, 98129545962855, 518149580437560
Offset: 0

Views

Author

Clark Kimberling, May 28 2000

Keywords

Links

Crossrefs

Cf. A055830.

Programs

  • Maple
    with(combinat);
T:= proc(n, k) option remember;
      if k<0 or k>n then 0
    elif k=0 then fibonacci(n+1)
    elif n=1 and k=1 then 0
    else T(n-1, k-1) + T(n-1, k) + T(n-2, k)
      fi; end:
seq(T(2*n+3, n), n=0..30); # G. C. Greubel, Jan 21 2020
  • Mathematica
    T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, Fibonacci[n+1], If[n==1 && k==1, 0, T[n-1, k-1] + T[n-1, k] + T[n-2, k]]]]; Table[T[2*n+3, n], {n,0,30}] (* G. C. Greubel, Jan 21 2020 *)
  • Sage
    @CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (k==0): return fibonacci(n+1)
    elif (n==1 and k==1): return 0
    else: return T(n-1, k-1) + T(n-1, k) + T(n-2, k)
[T(2*n+3, n) for n in (0..30)] # G. C. Greubel, Jan 21 2020

Formula

Conjecture: 5*n*(n+2)*(11*n-4)*a(n) +(-242*n^3-330*n^2+29*n+42)*a(n-1) -3*(3*n-1)*(11*n+7)*(3*n-2)*a(n-2)=0. - R. J. Mathar, Mar 13 2016

A055838 T(2n+4,n), where T is the array in A055830.

Original entry on oeis.org

5, 30, 162, 850, 4425, 22995, 119560, 622512, 3246750, 16963375, 88779900, 465386220, 2443204946, 12844119225, 67608235800, 356288599640, 1879625199825, 9925931817045, 52464942758250, 277546278287250
Offset: 0

Views

Author

Clark Kimberling, May 28 2000

Keywords

Links

Crossrefs

Cf. A055830.

Programs

  • Maple
    with(combinat);
T:= proc(n, k) option remember;
      if k<0 or k>n then 0
    elif k=0 then fibonacci(n+1)
    elif n=1 and k=1 then 0
    else T(n-1, k-1) + T(n-1, k) + T(n-2, k)
      fi; end:
seq(T(2*n+4, n), n=0..30); # G. C. Greubel, Jan 21 2020
  • Mathematica
    T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, Fibonacci[n+1], If[n==1 && k==1, 0, T[n-1, k-1] + T[n-1, k] + T[n-2, k]]]]; Table[T[2*n+4, n], {n,0,30}] (* G. C. Greubel, Jan 21 2020 *)
  • Sage
    @CachedFunction
def T(n, k):
    if (k<0 and k>n): return 0
    elif (k==0): return fibonacci(n+1)
    elif (n==1 and k==1): return 0
    else: return T(n-1, k-1) + T(n-1, k) + T(n-2, k)
[T(2*n+4, n) for n in (0..30)] # G. C. Greubel, Jan 21 2020

Formula

Conjecture: 5*n*(n+3)*(n-1)*a(n) -2*(n-1)*(11*n+8)*(n+2)*a(n-1) -3*(3*n-1)*(3*n-2)*(n+1)*a(n-2)=0. - R. J. Mathar, Mar 13 2016

A055839 T(2n+5,n), where T is the array in A055830.

Original entry on oeis.org

8, 58, 344, 1918, 10415, 55837, 297374, 1578160, 8359845, 44244825, 234094080, 1238598580, 6555004313, 34703385031, 183805639190, 973982775784, 5163655102685, 27389161216395, 145349642782140, 771718011707550
Offset: 0

Views

Author

Clark Kimberling, May 28 2000

Keywords

Links

Crossrefs

Cf. A055830.

Programs

  • Maple
    with(combinat);
T:= proc(n, k) option remember;
      if k<0 or k>n then 0
    elif k=0 then fibonacci(n+1)
    elif n=1 and k=1 then 0
    else T(n-1, k-1) + T(n-1, k) + T(n-2, k)
      fi; end:
seq(T(2*n+5, n), n=0..30); # G. C. Greubel, Jan 21 2020
  • Mathematica
    T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, Fibonacci[n+1], If[n==1 && k==1, 0, T[n-1, k-1] + T[n-1, k] + T[n-2, k]]]]; Table[T[2*n+5, n], {n,0,30}] (* G. C. Greubel, Jan 21 2020 *)
  • Sage
    @CachedFunction
def T(n, k):
    if (k<0 and k>n): return 0
    elif (k==0): return fibonacci(n+1)
    elif (n==1 and k==1): return 0
    else: return T(n-1, k-1) + T(n-1, k) + T(n-2, k)
[T(2*n+5, n) for n in (0..30)] # G. C. Greubel, Jan 21 2020

A055840 T(2n+6,n), where T is the array in A055830.

Original entry on oeis.org

13, 109, 707, 4184, 23720, 131389, 717927, 3889730, 20959485, 112529350, 602684170, 3222508015, 17211197614, 91855019053, 489986311295, 2612981923560, 13932202684630, 74280962031435, 396042187457445, 2111713236134025, 11260951929261216, 60058486994980518, 320362547860069042, 1709162928241695964
Offset: 0

Views

Author

Clark Kimberling, May 28 2000

Keywords

Links

Crossrefs

Cf. A055830.

Programs

  • Maple
    with(combinat);
T:= proc(n, k) option remember;
      if k<0 or k>n then 0
    elif k=0 then fibonacci(n+1)
    elif n=1 and k=1 then 0
    else T(n-1, k-1) + T(n-1, k) + T(n-2, k)
      fi; end:
seq(T(2*n+6, n), n=0..30); # G. C. Greubel, Jan 21 2020
  • Mathematica
    T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, Fibonacci[n+1], If[n==1 && k==1, 0, T[n-1, k-1] + T[n-1, k] + T[n-2, k]]]]; Table[T[2*(n+3), n], {n,0,30}] (* G. C. Greubel, Jan 21 2020 *)
  • Sage
    @CachedFunction
def T(n, k):
    if (k<0 and k>n): return 0
    elif (k==0): return fibonacci(n+1)
    elif (n==1 and k==1): return 0
    else: return T(n-1, k-1) + T(n-1, k) + T(n-2, k)
[T(2*n+6, n) for n in (0..30)] # G. C. Greubel, Jan 21 2020

Extensions

Terms a(19) onward added by G. C. Greubel, Jan 21 2020
Showing 1-10 of 30 results. Next

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