A163771 -id:A163771 - cp's OEIS frontend

A163774 Row sums of the central coefficients triangle (A163771).

1, 3, 13, 51, 201, 783, 3039, 11763, 45481, 175803, 679779, 2630367, 10187659, 39500373, 153329913, 595883763, 2318471289, 9030982491, 35216266947, 137469149451, 537152523711, 2100857828193, 8223917499477, 32219655346719, 126328429601451, 495676719721953, 1946227355491909

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. A056040, A163771.

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),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], {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)$, where i$ denotes the swinging factorial of i (A056040).

Conjecture: a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n+1,k)*binomial(2*k,k). - Werner Schulte, Nov 17 2015

More terms from Michel Marcus, Nov 24 2015

A163772 Triangle interpolating the swinging factorial (A056040) restricted to odd indices with its binomial inverse. Triangle read by rows. For n >= 0, k >= 0.

Original entry on oeis.org

1, 5, 6, 19, 24, 30, 67, 86, 110, 140, 227, 294, 380, 490, 630, 751, 978, 1272, 1652, 2142, 2772, 2445, 3196, 4174, 5446, 7098, 9240, 12012, 7869, 10314, 13510, 17684, 23130, 30228, 39468, 51480

Offset: 0

Peter Luschny, Aug 05 2009

			Triangle begins:
     1;
     5,    6;
    19,   24,   30;
    67,   86,  110,  140;
   227,  294,  380,  490,  630;
   751,  978, 1272, 1652, 2142, 2772;
  2445, 3196, 4174, 5446, 7098, 9240, 12012;

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 A163775. 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+1),n,true);

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[t[n, k], {n, 0, 7}, {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)*(2i+1)$ where i$ denotes the swinging factorial of i (A056040).

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).

A094796 Triangle read by rows giving coefficients of polynomials arising in successive differences of central binomial numbers.

Original entry on oeis.org

1, 3, 1, 9, 15, 6, 27, 108, 135, 42, 81, 594, 1539, 1530, 456, 243, 2835, 12555, 25245, 22122, 6120, 729, 12393, 83835, 281475, 482436, 383292, 101520, 2187, 51030, 489888, 2466450, 6916833, 10546200, 7786692, 1980720

Offset: 0

Benoit Cloitre, Jun 11 2004

Let D_0(n)=binomial(2*n,n) and D_{k+1}(n)=D_{k}(n+1)-D_{k}(n); then D_{k}(n)*(n+1)*(n+2)*...*(n+k) = binomial(2*n,n)*P_{k}(n) where P_{k} is a polynomial with integer coefficients of degree k.

			The third differences of the central binomial numbers are given by D_3(n) = binomial(2*n,n)*(n+1)*(n+2)*(n+3)*(27*n^3 + 108*n^2 + 135*n + 42) and the fourth row of the triangle is 27, 108, 135, 42.
From _M. F. Hasler_, Nov 15 2019: (Start)
The table reads:
  n  |  row(n)
  0  |    1
  1  |    3      1
  2  |    9     15       6
  3  |   27    108     135       42
  4  |   81    594    1539     1530      456
  5  |  243   2835   12555    25245    22122      6120
  6  |  729  12393   83835   281475   482436    383292    101520
  7  | 2187  51030  489888  2466450  6916833  10546200   7786692   1980720
  8  | 6561 201204 2602530 18329976 75981969 186899076 260520300 181218384 44634240
(End)

Cf. A000984 (central binomial coefficients), A163771 (square array of central binomial coefficients and higher differences), A000244 (column k=0).

Main diagonal gives A098461.

Dnk := proc(n,k)
    option remember;
    if k < 0 then
        0 ;
    elif k = 0 then
        binomial(2*n,n) ;
    else
        procname(n+1,k-1)-procname(n,k-1) ;
    end if;
end proc:
A094796 := proc(n,k)
    local xyvec,i,x ;
    xyvec := [] ;
    for i from 0 to n do
        xyvec := [op(xyvec),[i,Dnk(i,n)*mul(i+j,j=1..n)/Dnk(i,0)]] ;
    end do:
    CurveFitting[PolynomialInterpolation](xyvec,x) ;
    coeff(%,x,n-k) ;
end proc: # R. J. Mathar, Nov 19 2019

Dnk[n_, k_] := Dnk[n, k] = Which[k < 0, 0, k == 0, Binomial[2*n, n], True, Dnk[n + 1, k - 1] - Dnk[n, k - 1]];
T[n_, k_] := Module[{xyvec, i, x , ip}, xyvec = {}; For[i = 0, i <= n, i++, AppendTo[xyvec, {i, Dnk[i, n]*Product[i + j, {j, 1, n}]/Dnk[i, 0]}]]; ip = InterpolatingPolynomial[xyvec, x]; Coefficient[ip, x, n - k]];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 01 2024, after R. J. Mathar *)

apply( {A094796_row(n,D(n,k)=if(k,D(n+1,k-1)-D(n,k-1),binomial(2*n,n)))=Vec(polinterpolate([0..n],vector(n+1,k,D(k--,n)*(n+k)!/k!/binomial(2*k,k))))}, [0..8]) \\ M. F. Hasler, Nov 15 2019

T(n,0) = 3^n. T(n,1) = A027472(n+2) + 6*A027472(n+1). T(n,2) = 3*(2*A036217(n-2) + 15*A036217(n-3) + 18*A036217(n-4)). - R. J. Mathar, Nov 19 2019

Corrected and edited by M. F. Hasler, following observations by R. J. Mathar and Don Reble, Nov 15 2019

More terms from Don Reble, Nov 15 2019

A329533 First differences of A051924, or second differences of Central binomial coefficients A000984.

Original entry on oeis.org

3, 10, 36, 132, 490, 1836, 6930, 26312, 100386, 384540, 1478048, 5697720, 22019556, 85284920, 330961950, 1286562960, 5009003250, 19528599420, 76231136520, 297910080600, 1165429743660, 4563490674600, 17884841191620, 70148829799152, 275344923755700, 1081512966189656, 4250730282412320

Offset: 0

M. F. Hasler, Nov 15 2019

nonn

Cf. A000984, A051924, A163771, A094796.

Differences[#, 2] &@ Array[Binomial[2 #, #] &, 29, 0] (* Michael De Vlieger, Nov 15 2019 *)

C=vector(30,n,binomial(2*n--,n));C=C[^1]-C[^-1];C=C[^1]-C[^-1]

a(n) = A051924(n) - A051924(n-1) = A000984(n+2) - 2*A000984(n+1) + A000984(n).

a(n) = 3*(3*n+2)*(n+1)*binomial(2*n+4,n+2)/(4*(2*n+1)*(2*n+3)). - Alois P. Heinz, Sep 13 2024

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A163774 Row sums of the central coefficients triangle (A163771).

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A163772 Triangle interpolating the swinging factorial (A056040) restricted to odd indices with its binomial inverse. Triangle read by rows. For n >= 0, k >= 0.

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

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A094796 Triangle read by rows giving coefficients of polynomials arising in successive differences of central binomial numbers.

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A329533 First differences of A051924, or second differences of Central binomial coefficients A000984.

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