A050935 -id:A050935 - cp's OEIS frontend

A124304 Riordan array (1, x*(1-x^2)).

1, 0, 1, 0, 0, 1, 0, -1, 0, 1, 0, 0, -2, 0, 1, 0, 0, 0, -3, 0, 1, 0, 0, 1, 0, -4, 0, 1, 0, 0, 0, 3, 0, -5, 0, 1, 0, 0, 0, 0, 6, 0, -6, 0, 1, 0, 0, 0, -1, 0, 10, 0, -7, 0, 1, 0, 0, 0, 0, -4, 0, 15, 0, -8, 0, 1, 0, 0, 0, 0, 0, -10, 0, 21, 0, -9, 0, 1, 0, 0, 0, 0, 1, 0, -20, 0, 28, 0, -10, 0, 1

Offset: 0

Paul Barry, Oct 25 2006

T(2n,n) is a signed aerated version of C(2n,n).

Inverse is A124305.

			Triangle begins
  1;
  0,  1;
  0,  0,  1;
  0, -1,  0,  1;
  0,  0, -2,  0,  1;
  0,  0,  0, -3,  0,  1;
  0,  0,  1,  0, -4,  0,  1;
  0,  0,  0,  3,  0, -5,  0,  1;
  0,  0,  0,  0,  6,  0, -6,  0,  1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.

Cf. A014021 (diagonal sums), A050935 (row sums), A124305 (inverse).

Cf. A002054, A079977, A126869, A138364, A176971.

A124304:= func< n,k | (&+[(-1)^j*Binomial(k,k-j)*Binomial(k,n-k-j) : j in [0..n]]) >;
[A124304(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Aug 18 2023

A124304[n_, k_]:= Binomial[k, (n-k)/2]*(-1)^((n-k)/2)*(1+(-1)^(n-k))/2;
Table[A124304[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 18 2023 *)

def A124304(n, k): return binomial(k, (n-k)//2)*(-1)^((n-k)//2)*(1+(-1)^(n-k))/2
flatten([[A124304(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Aug 18 2023

T(n, k) = Sum_{j=0..n} C(k,k-j)*C(k,n-k-j)*(-1)^j.
T(n, k) = C(k,(n-k)/2)*(-1)^((n-k)/2)*(1 + (-1)^(n-k))/2.
Sum_{k=0..n} T(n, k) = A050935(n+2).
Sum_{k=0..floor(n/2)} T(n-k, k) = A014021(n).
T(2*n, n) = (1 - 2*0^(n+2 mod 4))*A126869(n).
From G. C. Greubel, Aug 18 2023: (Start)
T(2*n-1, n-1) = (1 - 2*0^(n+1 mod 4))*A138364(n-1).
T(2*n-1, n+1) = (1 - 2*0^(n mod 4))*((1+(-1)^n)/2)*A002054(floor(n/2)).
Sum_{k=0..n} (-1)^k*T(n, k) = A176971(n+3).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (1 - 2*0^(n+2 mod 4))*A079977(n).
G.f.: 1/(1 - x*y*(1-x^2)). (End)

A126030 Riordan array (1/(1+x^3),x/(1+x^3)).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, -1, 0, 0, 1, 0, -2, 0, 0, 1, 0, 0, -3, 0, 0, 1, 1, 0, 0, -4, 0, 0, 1, 0, 3, 0, 0, -5, 0, 0, 1, 0, 0, 6, 0, 0, -6, 0, 0, 1, -1, 0, 0, 10, 0, 0, -7, 0, 0, 1, 0, -4, 0, 0, 15, 0, 0

Offset: 0

Paul Barry, Dec 15 2006

Inverse is A111373. Row sums are A050935(n+2). Diagonal sums are an alternating sign version of A000931(n+3) with g.f. 1/(1-x^2+x^3).

			Triangle begins
.1,
.0, 1,
.0, 0, 1,
.-1, 0, 0, 1,
.0, -2, 0, 0, 1,
.0, 0, -3, 0, 0, 1,
.1, 0, 0, -4, 0, 0, 1,
.0, 3, 0, 0, -5, 0, 0, 1,
.0, 0, 6, 0, 0, -6, 0, 0, 1,
.-1, 0, 0, 10, 0, 0, -7, 0, 0, 1,
.0, -4, 0, 0, 15, 0, 0, -8, 0, 0, 1

Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3

Number triangle T(n,k)=C(k+(n-k)/3,(n-k)/3)*(-1)^(n-k)*(2*cos(2*pi*(n-k)/3)+1)/3

A109582 Expansion of e.g.f.: -1/(1+x-x^3).

Original entry on oeis.org

-1, 1, -2, 0, 24, -240, 1440, -5040, -40320, 1088640, -14515200, 119750400, 0, -24908083200, 610248038400, -9153720576000, 62768369664000, 1422749712384000, -70426110763008000, 1703031405723648000, -24329020081766400000, -51090942171709440000

Offset: 0

Roger L. Bagula, Jun 29 2005

sign

Cf. A050935, A104769.

g:=-1/(1+x-x^3):gser:=series(g,x=0,26): -1,seq(n!*coeff(gser,x^n),n=1..22);

E.g.f.: -1/(1+x-x^3).
a(n) = n!*A104769(n+1). - R. J. Mathar, Aug 20 2021
D-finite with recurrence a(n) +n*a(n-1) -n*(n-1)*(n-2)*a(n-3) = 0. - R. J. Mathar, Aug 20 2021

A121230 First Hadamard-Sylvester matrix self -similar matrix based on the Padovan/ Minimal Pisot 3 X 3 matrix as an 9 X 9 matrix: Characteristic Polynomial:1 - x - x^3 - x^4 - x^5 + 3 x^6 + 2 x^7 - x^9.

Original entry on oeis.org

0, 13, 5, 22, 42, 54, 126, 192, 347, 631, 1056, 1914, 3320, 5814, 10276, 17921, 31549, 55338, 97026, 170454, 298914, 524684, 920815, 1615647, 2835660, 4975898, 8732160, 15324202, 26891432, 47191909, 82815621, 145331022, 255039162

Offset: 1

Roger L. Bagula, Aug 13 2006

As far as I can tell by searching the Internet, this matrix and this approach to sequences is entirely new and unique. The second of these matrices at 81 X 81 gives a new fractal that is Cantor dust like. aa = Table[M[[n, m]]*M[[i, j]], {n, 1, 9 }, {m, 1, 9}, {i, 1, 9}, {j, 1, 9}]; M2 = Flatten[Table[{Flatten[Table[aa[[ n, m]][[1, i]], {n, 1, 9}, {i, 1, 9}]], Flatten[Table[aa[[n, m]][[2, i]], {n, 1, 9}, {i, 1, 9}]], Flatten[Table[aa[[ n, m]][[3, i]], {n, 1, 9}, {i, 1, 9}]], Flatten[Table[aa[[ n, m]][[4, i]], {n, 1, 9}, {i, 1, 9}]], Flatten[Table[aa[[ n, m]][[5, i]], {n, 1, 9}, {i, 1, 9}]], Flatten[Table[aa[[n, m]][[6, i]], {n, 1, 9}, {i, 1, 9}]], Flatten[Table[aa[[ n, m]][[7, i]], {n, 1, 9}, {i, 1, 9}]], Flatten[Table[aa[[ n, m]][[8, i]], {n, 1, 9}, {i, 1, 9}]], Flatten[Table[aa[[ n, m]][[9, i]], {n, 1, 9}, {i, 1, 9}]]}, {m, 1, 9}], 1]; ListDensityPlot[M2, Mesh -> False]

Index entries for linear recurrences with constant coefficients, signature (1, 1, 1, -1, 1, -1).

Cf. A000931.

Clear[t, M, a, v, a0] t[n_, m_] := {{0, 1, 0}, {0, 0, 1}, {1, 1, 0}}[[n, m]] a0 = Table[t[n, m]*t[i, j], {n, 1, 3}, {m, 1, 3}, {i, 1, 3}, {j, 1, 3}]; M = Flatten[Table[{Flatten[Table[a0[[ n, m]][[1, i]], {n, 1, 3}, {i, 1, 3}]], Flatten[Table[a0[[n, m]][[2, i]], {n, 1, 3}, {i, 1, 3}]], Flatten[Table[a0[[n, m]][[3, i]], {n, 1, 3}, {i, 1, 3}]]}, {m, 1, 3}], 1] v[1] = Table[Fibonacci[n], {n, 0, 8}] v[n_] := v[n] = M.v[n - 1] a = Table[Floor[v[n][[1]]], {n, 1, 50}] Det[M - x*IdentityMatrix[9]] Factor[%] aaa = Table[x /. NSolve[Det[M - x*IdentityMatrix[9]] == 0, x][[n]], {n, 1, 9}] Abs[aaa] a1 = Table[N[a[[n]]/a[[n - 1]]], {n, 7, 50}] ListDensityPlot[M, Mesh -> False]

M={{0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1}, { 0, 0, 0, 0, 0, 0, 1, 1, 0}, {0, 1, 0, 0, 0, 0, 0, 1, 0}, { 0, 0, 1, 0, 0, 0, 0, 0, 1}, {1, 1, 0, 0, 0, 0, 1, 1, 0}, { 0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, { 0, 0, 0, 1, 1, 0, 0, 0, 0}} v[1] = Table[Fibonacci[n], {n, 0, 8}] v[n_] := v[n] = M.v[n - 1] a(n) = v[n][[1]]

G.f.: x^2(13-8x+4x^2+2x^3-2x^4)/((1-2x+x^2-x^3)(1+x-x^3)). a(n) = a(n-1) +a(n-2) +a(n-3) -a(n-4) +a(n-5) -a(n-6). Partial fraction decomposition yield decomposition in terms of A005314 and A050935. [From R. J. Mathar, Nov 26 2008]

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A124304 Riordan array (1, x*(1-x^2)).

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A126030 Riordan array (1/(1+x^3),x/(1+x^3)).

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A109582 Expansion of e.g.f.: -1/(1+x-x^3).

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A121230 First Hadamard-Sylvester matrix self -similar matrix based on the Padovan/ Minimal Pisot 3 X 3 matrix as an 9 X 9 matrix: Characteristic Polynomial:1 - x - x^3 - x^4 - x^5 + 3 x^6 + 2 x^7 - x^9.

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