A095704 -id:A095704 - cp's OEIS frontend

A238801 Triangle T(n,k), read by rows, given by T(n,k) = C(n+1, k+1)*(1-(k mod 2)).

1, 2, 0, 3, 0, 1, 4, 0, 4, 0, 5, 0, 10, 0, 1, 6, 0, 20, 0, 6, 0, 7, 0, 35, 0, 21, 0, 1, 8, 0, 56, 0, 56, 0, 8, 0, 9, 0, 84, 0, 126, 0, 36, 0, 1, 10, 0, 120, 0, 252, 0, 120, 0, 10, 0, 11, 0, 165, 0, 462, 0, 330, 0, 55, 0, 1, 12, 0, 220, 0, 792, 0, 792, 0, 220, 0, 12, 0

Offset: 0

Philippe Deléham, Mar 05 2014

Row sums are powers of 2.

			Triangle begins:
1;
2, 0;
3, 0, 1;
4, 0, 4, 0;
5, 0, 10, 0, 1;
6, 0, 20, 0, 6, 0;
7, 0, 35, 0, 21, 0, 1;
8, 0, 56, 0, 56, 0, 8, 0;
9, 0, 84, 0, 126, 0, 36, 0, 1;
10, 0, 120, 0, 252, 0, 120, 0, 10, 0; etc.

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. Columns: A000027, A000292, A000389, A000580, A000582, A001288, A010966, A010968, A010970, A010972, A010974, A010976, A010980, A010982, A010984, A010986, A010988, A010990, A010992, A010994, A010996, A010998, A011000, A017713, A017715, A017717, A017719, A017721, A017723, A017725, A017727, A017729, A017731, A017733, A017735, A017737, A017739, A017741, A017743, A017745, A017747, A017749, A017751, A017753, A017755, A017757, A017759, A017761, A017763.

Cf. A095704, A178616.

Table[Binomial[n + 1, k + 1]*(1 - Mod[k , 2]), {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Nov 22 2017 *)

T(n,k) = binomial(n+1, k+1)*(1-(k % 2));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Nov 23 2017

G.f.: 1/((1+(y-1)*x)*(1-(y+1)*x)).
T(n,k) = 2*T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = 1, T(1,0) = 2, T(1,1) = 0, T(n,k) = 0 if k<0 or if k>n.
Sum_{k=0..n} T(n,k)*x^k = A000027(n+1), A000079(n), A015518(n+1), A003683(n+1), A079773(n+1), A051958(n+1), A080920(n+1), A053455(n), A160958(n+1) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively.

A178616 Triangle by columns, odd columns of Pascal's triangle A007318, otherwise (1, 0, 0, 0, ...).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 0, 1, 0, 4, 0, 4, 1, 0, 5, 0, 10, 0, 1, 0, 6, 0, 20, 0, 6, 1, 0, 7, 0, 35, 0, 21, 0, 1, 0, 8, 0, 56, 0, 56, 0, 8, 0, 1, 0, 9, 0, 84, 0, 126, 0, 36, 0, 1, 0, 10, 0, 120, 0, 252, 0, 120, 0, 10, 1

Offset: 0

Gary W. Adamson, May 30 2010

Row sums = a variant of A052950, starting (1, 1, 3, 4, 9, 16, 33, ...); whereas A052950 starts (2, 1, 3, 4, 9, ...).

Column 1 of the inverse of A178616 is a signed variant of A065619 prefaced with a 0; where A065619 = (1, 2, 3, 8, 25, 96, 427, ...).

			First few rows of the triangle:
  1,
  0,  1;
  0,  2, 1;
  0,  3, 0,   1
  0,  4, 0,   4, 1;
  0,  5, 0,  10, 0,   1;
  0,  6, 0,  20, 0,   6, 1;
  0,  7, 0,  35, 0,  21, 0,   1;
  0,  8, 0,  56, 0,  56, 0,   8, 1;
  0,  9, 0,  84, 0, 126, 0,  36, 0,  1;
  0, 10, 0, 120, 0, 252, 0, 120, 0, 10, 1;
  0, 11, 0, 165, 0, 462, 0, 330, 0, 55, 0, 1;
  ...

Cf. A162109, A065619, A052950, A095704.

Triangle, odd columns of Pascal's triangle; (1, 0, 0, 0, ...) as even columns k.
Alternatively, (since A178616 + A162169 - Identity matrix) = Pascal's triangle,
we can begin with Pascal's triangle, subtract A162169, then add the Identity
matrix to obtain A178616.

A096754 Triangle read by rows giving coefficients of the trigonometric expansion of Cos(n*x).

Original entry on oeis.org

1, 1, 0, -1, 1, 0, -3, 1, 0, -6, 0, 1, 1, 0, -10, 0, 5, 1, 0, -15, 0, 15, 0, -1, 1, 0, -21, 0, 35, 0, -7, 1, 0, -28, 0, 70, 0, -28, 0, 1, 1, 0, -36, 0, 126, 0, -84, 0, 9, 1, 0, -45, 0, 210, 0, -210, 0, 45, 0, -1, 1, 0, -55, 0, 330, 0, -462, 0, 165, 0, -11, 1, 0, -66, 0, 495, 0, -924, 0, 495, 0, -66, 0, 1, 1, 0, -78, 0, 715

Offset: 1

Robert G. Wilson v, Jul 07 2004

T(n,k)=cos(n,k)*cos(pi*k/2) begins {1}, {1,0}, {1,0,-1}, {1,0,-3,0},... - Paul Barry, May 21 2006

			The trigonometric expansion of Cos(4x) = Cos[x]^4 - 6*Cos[x]^2*Sin[x]^2 + Sin[x]^4, therefore the fourth row is 1, 0, -6, 0, 1.
The trigonometric expansion of Cos(5x) = Cos[x]^5 - 10*Cos[x]^3*Sin[x]^2 + 5*Cos[x]*Sin[x]^4, therefore the fifth row of the triangle is 1, 0, -10, 0, 5
The table begins:
1
1 0 -1
1 0 -3
1 0 -6 0 1
1 0 -10 0 5
1 0 -15 0 15 0 -1
1 0 -21 0 35 0 -7
1 0 -28 0 70 0 -28 0 1

Clark Kimberling, Polynomials associated with reciprocation, JIS 12 (2009) 09.3.4, section 5.

Another version of the triangle in A034839. Cf. A095704.
First column is A000012 = C(n, 0), third column is A000217 = C(n, 2), fifth column is A000332 = C(n, 4), seventh column is A000579 = C(n, 6), ninth column is A000581 = C(n, 8).
A001287 = C(n, 10), A010965 = C(n, 12), A010967 = C(n, 14), A010969 = C(n, 16), A010971 = C(n, 18),
A010973 = C(n, 20), A010975 = C(n, 22), A010977 = C(n, 24), A010979 = C(n, 26), A010981 = C(n, 28),
A010983 = C(n, 30), A010985 = C(n, 32), A010987 = C(n, 34), A010989 = C(n, 36), A010991 = C(n, 38),
A010993 = C(n, 40), A010995 = C(n, 42), A010997 = C(n, 44), A010999 = C(n, 46), A011001 = C(n, 48),
A017714 = C(n, 50), A017716 = C(n, 52), A017718 = C(n, 54), A017720 = C(n, 56), etc.

Flatten[Table[ Plus @@ CoefficientList[ TrigExpand[ Cos[n*x]], { Cos[x], Sin[x]}], {n, 13}]]

A135685 Triangular sequence of the coefficients of the numerator of the rational recursive sequence for tan(n*y) with x = tan(y).

Original entry on oeis.org

0, 0, 1, 0, -2, 0, -3, 0, 1, 0, 4, 0, -4, 0, 5, 0, -10, 0, 1, 0, -6, 0, 20, 0, -6, 0, -7, 0, 35, 0, -21, 0, 1, 0, 8, 0, -56, 0, 56, 0, -8, 0, 9, 0, -84, 0, 126, 0, -36, 0, 1, 0, -10, 0, 120, 0, -252, 0, 120, 0, -10, 0, -11, 0, 165, 0, -462, 0, 330, 0, -55, 0, 1

Offset: 0

Roger L. Bagula, Feb 17 2008

Signed version of A034867 with interlaced zeros. - Joerg Arndt, Sep 14 2014

The negatives of these terms gives the coefficients for the numerators for when n is negative (i.e. tan(-n*y) = -tan(n*y)). - James Burling, Sep 14 2014

			Triangle starts:
  0;
  0,   1;
  0,  -2;
  0,  -3,  0,   1;
  0,   4,  0,  -4;
  0,   5,  0, -10,  0,    1;
  0,  -6,  0,  20,  0,   -6;
  0,  -7,  0,  35,  0,  -21,  0,   1;
  0,   8,  0, -56,  0,   56,  0,  -8;
  0,   9,  0, -84,  0,  126,  0, -36,  0,   1;
  0, -10,  0, 120,  0, -252,  0, 120,  0, -10;
  0, -11,  0, 165,  0, -462,  0, 330,  0, -55,  0,  1;

Robert Israel, Table of n, a(n) for n = 0..10082
Clark Kimberling, Polynomials associated with reciprocation, JIS 12 (2009) 09.3.4, section 5.

Cf. A095704, A162590.

g[0]:= 0:
g[1]:= x;
for n from 2 to 20 do
g[n]:= expand(-2*(-1)^n*g[n-1]+(x^2+1)*g[n-2])
od:
0, seq(seq(coeff(g[n],x,j),j=0..degree(g[n])),n=1..20); # Robert Israel, Sep 14 2014

p[n_, x_]:= p[n, x]= If[n<2, n*x, (p[n-1, x] + x)/(1 - x*p[n-1, x])];
Table[CoefficientList[Numerator[FullSimplify[p[n, x]]], x], {n,0,12}]//Flatten

def p(n, x): return n*x if (n<2) else 2*(-1)^(n+1)*p(n-1,x) + (1+x^2)*p(n-2,x)
def A135685(n,k): return ( p(n,x) ).series(x,n+1).list()[k]
flatten([[A135685(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Nov 26 2021

p(n, x) = (p(n-1, x) + x)/(1 - x*p(n-1, x)), with p(0, x) = 0, p(1, x) = x.

Sum_{j} T(n,j)*x^j = g(n,x) where g(0,x) = 0, g(1,x) = x, g(n,x) = -2*(-1)^n*g(n-1,x) + (x^2+1)*g(n-2,x). - Robert Israel, Sep 14 2014

Prepended first term and offset corrected by James Burling, Sep 14 2014

cp's OEIS Frontend

A238801 Triangle T(n,k), read by rows, given by T(n,k) = C(n+1, k+1)*(1-(k mod 2)).

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A178616 Triangle by columns, odd columns of Pascal's triangle A007318, otherwise (1, 0, 0, 0, ...).

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A096754 Triangle read by rows giving coefficients of the trigonometric expansion of Cos(n*x).

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A135685 Triangular sequence of the coefficients of the numerator of the rational recursive sequence for tan(n*y) with x = tan(y).

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