A163772 -id:A163772 - cp's OEIS frontend

A163775 Row sums of triangle A163772.

1, 11, 73, 403, 2021, 9567, 43611, 193683, 844213, 3629083, 15437951, 65143503, 273148279, 1139548469, 4734740493, 19606960755, 80969809797, 333601494651

Offset: 0

Peter Luschny, Aug 05 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.

Cf. A163772.

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((-1)^(n-i)*binomial(n-k,n-i)*swing(2*i+1),i=k..n),k=0..n) end:

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[(-1)^(n - i)* Binomial[n - k, n - i]*sf[2*i + 1], {i, k, n}]; Table[Sum[t[n, k], {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Aug 04 2017 *)

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

A163771 Triangle interpolating the swinging factorial (A056040) restricted to even indices with its binomial inverse. Same as interpolating the central trinomial coefficients (A002426) with the central binomial coefficients (A000984).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 7, 10, 14, 20, 19, 26, 36, 50, 70, 51, 70, 96, 132, 182, 252, 141, 192, 262, 358, 490, 672, 924, 393, 534, 726, 988, 1346, 1836, 2508, 3432, 1107, 1500, 2034, 2760, 3748, 5094, 6930, 9438, 12870

Offset: 0

Peter Luschny, Aug 05 2009

Triangle read by rows. For n >= 0, k >= 0 let T(n,k) = Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*(2i)$ where i$ denotes the swinging factorial of i (A056040).

This is also the square array of central binomial coefficients A000984 in column 0 and higher (first: A051924, second, etc.) differences in subsequent columns, read by antidiagonals. - M. F. Hasler, Nov 15 2019

			Triangle begins
    1;
    1,   2;
    3,   4,   6;
    7,  10,  14,  20;
   19,  26,  36,  50,  70;
   51,  70,  96, 132, 182, 252;
  141, 192, 262, 358, 490, 672, 924;
From _M. F. Hasler_, Nov 15 2019: (Start)
The square array having central binomial coefficients A000984 in column 0 and higher differences in subsequent columns (col. 1 = A051924) starts:
     1   1    3    7    19    51 ...
     2   4   10   26    70   192 ...
     6  14   36   96   262   726 ...
    20  50  132  358   988  2760 ...
    70 182  490 1346  3748 10540 ...
   252 672 1836 5094 14288 40404 ...
  (...)
Read by falling antidiagonals this yields the same sequence. (End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.
M. Z. Spivey and L. L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.

Row sums are A163774. Cf. A056040, A163650, A163771, A163772, A002426, A000984.

For the functions 'DiffTria' and 'swing' see A163770. Computes n rows of the triangle.
a := n -> DiffTria(k->swing(2*k),n,true);

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[(-1)^(n - i)*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 *)

A163872 Inverse binomial transform of the beta numbers 1/beta(n+1,n+1) (A002457).

Original entry on oeis.org

1, 5, 19, 67, 227, 751, 2445, 7869, 25107, 79567, 250793, 786985, 2460397, 7667921, 23832931, 73902627, 228692115, 706407903, 2178511449, 6708684009, 20632428249, 63380014845, 194486530791, 596213956023, 1826103432573, 5588435470401, 17089296473655

Offset: 0

Peter Luschny, Aug 06 2009

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

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.

Cf. A163772.

a := proc(n) local i; add((-1)^(n-i)*binomial(n,i)/Beta(i+1,i+1),i=0..n) end:
seq(simplify((-1)^n*hypergeom([-n,3/2], [1], 4)),n=0..26); # Peter Luschny, Apr 26 2016

CoefficientList[Series[Sqrt[x+1]/(1-3*x)^(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[(-1)^(n-i)*Binomial[n, n-i]*sf[2*i+1], {i, 0, n}]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jul 26 2013 *)

O.g.f.: A(x)=1/(1-x*M(x))^3, M(x) - o.g.f. of A001006. a(n) = sum(k^3/n *sum(C(n,j)*C(j,2*j-n-k), j=0..n), k=1..n). - Vladimir Kruchinin, Sep 06 2010
Recurrence: n*a(n) = (2*n+3)*a(n-1) + 3*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 21 2012
a(n) ~ 4*3^(n-1/2)*sqrt(n)/sqrt(Pi). - Vaclav Kotesovec, Oct 21 2012
a(n) = (-1)^n*hypergeom([-n,3/2], [1], 4). - Peter Luschny, Apr 26 2016
From Seiichi Manyama, Aug 22 2025: (Start)
a(n) = (-1/4)^n * Sum_{k=0..n} (-3)^k * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} (-1)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 3^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n,n-k). (End)

A163770 Triangle read by rows interpolating the swinging subfactorial (A163650) with the swinging factorial (A056040).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 4, 6, -9, -7, -4, 0, 6, 44, 35, 28, 24, 24, 30, -165, -121, -86, -58, -34, -10, 20, 594, 429, 308, 222, 164, 130, 120, 140, -2037, -1443, -1014, -706, -484, -320, -190, -70, 70, 6824, 4787, 3344, 2330, 1624, 1140, 820, 630, 560, 630

Offset: 0

Peter Luschny, Aug 05 2009

An analog to the derangement triangle (A068106).

			1
0, 1
1, 1, 2
2, 3, 4, 6
-9, -7, -4, 0, 6
44, 35, 28, 24, 24, 30
-165, -121, -86, -58, -34, -10, 20

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.
M. Z. Spivey and L. L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.

Row sums are A163773.

Cf. A056040, A163650, A163771, A163772, A068106.

DiffTria := 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,(-1)^(m-i)*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.
A163770 := n -> DiffTria(k->swing(k),n,true);
A068106 := n -> DiffTria(k->factorial(k),n,true);

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[(-1)^(n - i)*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 *)

T(n,k) = Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*i$ where i$ denotes the swinging factorial of i (A056040).

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A163775 Row sums of triangle A163772.

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A163771 Triangle interpolating the swinging factorial (A056040) restricted to even indices with its binomial inverse. Same as interpolating the central trinomial coefficients (A002426) with the central binomial coefficients (A000984).

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A163872 Inverse binomial transform of the beta numbers 1/beta(n+1,n+1) (A002457).

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A163770 Triangle read by rows interpolating the swinging subfactorial (A163650) with the swinging factorial (A056040).

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