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.

A163845 Row sums of triangle A163842.

Original entry on oeis.org

1, 13, 109, 765, 4881, 29369, 169919, 956237, 5272945, 28632525, 153638211, 816715073, 4309138419, 22598433555, 117926579385, 612863125965, 3174156512865, 16392351740045, 84448387609475, 434142126555125, 2227861180841895, 11414655603043335, 58403793025471605
Offset: 0

Views

Author

Peter Luschny, Aug 06 2009

Keywords

Links

Crossrefs

Cf. A056040, A163842.

Programs

  • Maple
    swing := proc(n) option remember; if n = 0 then 1 elif
irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
a := proc(n) local i,k; add(add(binomial(n-k,n-i)*swing(2*i+1),i=k..n),k=0..n) end:
  • Mathematica
    swing[n_] := n! / Floor[n/2]!^2; a[n_] := Sum[Binomial[n-k, n-i] * swing[2*i+1], {k, 0, n}, {i, k, n}]; Array[a, 30, 0] (* Amiram Eldar, Aug 22 2025 *)

Formula

a(n) = Sum_{k=0..n} Sum_{i=k..n} binomial(n-k,n-i)*(2i+1)$ where i$ denotes the swinging factorial of i (A056040).

A163869 Binomial transform of the beta numbers 1/beta(n+1,n+1) (A002457).

Original entry on oeis.org

1, 7, 43, 249, 1395, 7653, 41381, 221399, 1175027, 6196725, 32512401, 169863147, 884318973, 4589954619, 23761814955, 122735222505, 632698778835, 3255832730565, 16728131746145, 85826852897675, 439793834236745, 2251006269442815, 11509340056410735, 58790764269668805
Offset: 0

Views

Author

Peter Luschny, Aug 06 2009

Keywords

Comments

Also a(n) = Sum_{i=0..n} binomial(n,n-i) (2*i+1)$ where i$ denotes the swinging factorial of i (A056040).

Links

Crossrefs

Cf. A163842, A163872, A387208.

Programs

  • Maple
    a := proc(n) local i; add(binomial(n,i)/Beta(i+1,i+1), i=0..n) end:
  • Mathematica
    CoefficientList[Series[-Sqrt[x-1]/(5*x-1)^(3/2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 21 2012 *)
sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[n_] := Sum[ Binomial[n, n-i]*sf[2*i+1], {i, 0, n}]; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Jul 26 2013 *)
Table[Hypergeometric2F1[3/2, -n, 1, -4], {n, 0, 20}] (* Vladimir Reshetnikov, Apr 25 2016 *)

Formula

From Vaclav Kotesovec, Oct 21 2012: (Start)
G.f.: -sqrt(x-1)/(5*x-1)^(3/2).
Recurrence: n*a(n) = (6*n+1)*a(n-1) - 5*(n-1)*a(n-2).
a(n) ~ 4*5^(n-1/2)*sqrt(n)/sqrt(Pi).
(End)
a(n) = hypergeom([3/2, -n], [1], -4) = hypergeom([3/2, n+1], [1], 4/5)/(5*sqrt(5)). - Vladimir Reshetnikov, Apr 25 2016
E.g.f.: exp(3*x) * ((1 + 4*x) * BesselI(0,2*x) + 4 * x * BesselI(1,2*x)). - Ilya Gutkovskiy, Nov 19 2021
From Seiichi Manyama, Aug 22 2025: (Start)
a(n) = (1/4)^n * Sum_{k=0..n} 5^k * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(n,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 5^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n,n-k). (End)

A163840 Triangle interpolating the binomial transform of the swinging factorial (A163865) with the swinging factorial (A056040).

Original entry on oeis.org

1, 2, 1, 5, 3, 2, 16, 11, 8, 6, 47, 31, 20, 12, 6, 146, 99, 68, 48, 36, 30, 447, 301, 202, 134, 86, 50, 20, 1380, 933, 632, 430, 296, 210, 160, 140, 4251, 2871, 1938, 1306, 876, 580, 370, 210, 70, 13102, 8851, 5980, 4042, 2736, 1860, 1280, 910, 700, 630
Offset: 0

Views

Author

Peter Luschny, Aug 06 2009

Keywords

Comments

Triangle read by rows.
An analog to the binomial triangle of the factorials (A076571).

Examples

			Triangle begins
    1;
    2,   1;
    5,   3,   2;
   16,  11,   8,   6;
   47,  31,  20,  12,  6;
  146,  99,  68,  48, 36, 30;
  447, 301, 202, 134, 86, 50, 20;

Links

Crossrefs

Row sums are A163843.
Cf. A056040, A163865, A163841, A163842, A163650.

Programs

  • Maple
    SumTria := proc(f,n,display) local m,A,j,i,T; T:=f(0);
for m from 0 by 1 to n-1 do A[m] := f(m);
for j from m by -1 to 1 do A[j-1] := A[j-1] + A[j] od;
for i from 0 to m do T := T,A[i] od;
if display then print(seq(T[i],i=nops([T])-m..nops([T]))) fi;
od; subsop(1=NULL,[T]) end:
swing := proc(n) option remember; if n = 0 then 1 elif
irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
# Computes n rows of the triangle:
A163840 := n -> SumTria(swing,n,true);
  • Mathematica
    sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[Binomial[n - k, n - i]*sf[i], {i, k, n}]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)

Formula

T(n,k) = Sum_{i=k..n} binomial(n-k,n-i)*i$ where i$ denotes the swinging factorial of i (A056040), for n >= 0, k >= 0.

A163841 Triangle interpolating the swinging factorial (A056040) restricted to even indices with its binomial transform. Same as interpolating bilateral Schroeder paths (A026375) with the central binomial coefficients (A000984).

Original entry on oeis.org

1, 3, 2, 11, 8, 6, 45, 34, 26, 20, 195, 150, 116, 90, 70, 873, 678, 528, 412, 322, 252, 3989, 3116, 2438, 1910, 1498, 1176, 924, 18483, 14494, 11378, 8940, 7030, 5532, 4356, 3432, 86515, 68032, 53538
Offset: 0

Views

Author

Peter Luschny, Aug 06 2009

Keywords

Comments

For n >= 0, k >= 0 let T(n,k) = sum{i=k..n} binomial(n-k,n-i)*(2i)$ where i$ denotes the swinging factorial of i (A056040). Triangle read by rows.

Examples

			Triangle begins
     1;
     3,    2;
    11,    8,    6;
    45,   34,   26,   20;
   195,  150,  116,   90,   70;
   873,  678,  528,  412,  322,  252;
  3989, 3116, 2438, 1910, 1498, 1176,  924;

Links

Crossrefs

Row sums are A163844. Cf. A056040, A163650, A163841, A163842, A163840, A026375, A002426, A000984.

Programs

  • Maple
    Computes n rows of the triangle. For the functions 'SumTria' and 'swing' see A163840.
a := n -> SumTria(k->swing(2*k),n,true);
  • Mathematica
    sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[Binomial[n - k, n - i]*sf[2*i], {i, k, n}]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)
Showing 1-4 of 4 results.

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