A034867 -id:A034867 - cp's OEIS frontend

A109447 Binomial coefficients C(n,k) with n-k odd, read by rows.

1, 2, 1, 3, 4, 4, 1, 10, 5, 6, 20, 6, 1, 21, 35, 7, 8, 56, 56, 8, 1, 36, 126, 84, 9, 10, 120, 252, 120, 10, 1, 55, 330, 462, 165, 11, 12, 220, 792, 792, 220, 12, 1, 78, 715, 1716, 1287, 286, 13, 14, 364, 2002, 3432, 2002, 364, 14, 1, 105, 1365, 5005, 6435, 3003, 455, 15

Offset: 1

Philippe Deléham, Aug 27 2005

The same as A119900 without 0's. A reflected version of A034867 or A202064. - Alois P. Heinz, Feb 07 2014
From Vladimir Shevelev, Feb 07 2014: (Start)
Also table of coefficients of polynomials  P_1(x)=1, P_2(x)=2, for n>=2, P_(n+1)(x) = 2*P_n(x)+(x-1)* P_(n-1)(x).  The polynomials P_n(x)/2^(n-1) are connected with sequences A000045 (x=5), A001045 (x=9), A006130 (x=13), A006131 (x=17), A015440 (x=21), A015441 (x=25), A015442 (x=29), A015443 (x=33), A015445 (x=37), A015446 (x=41), A015447 (x=45), A053404 (x=49); also the polynomials P_n(x) are connected with sequences A000129, A002605, A015518, A063727, A085449, A002532, A083099, A015519, A003683, A002534, A083102, A015520. (End)

			Starred terms in Pascal's triangle (A007318), read by rows:
1;
1*, 1;
1, 2*, 1;
1*, 3, 3*, 1;
1, 4*, 6, 4*, 1;
1*, 5, 10*, 10, 5*, 1;
1, 6*, 15, 20*, 15, 6*, 1;
1*, 7, 21*, 35, 35*, 21, 7*, 1;
1, 8*, 28, 56*, 70, 56*, 28, 8*, 1;
1*, 9, 36*, 84, 126*, 126, 84*, 36, 9*, 1;
Triangle T(n,k) begins:
1;
2;
1,    3;
4,    4;
1,   10,  5;
6,   20,  6;
1,   21,  35,   7;
8,   56,  56,   8;
1,   36, 126,  84,  9;
10, 120, 252, 120, 10;

Alois P. Heinz, Rows n = 1..200, flattened

Cf. A109446.

T:= (n, k)-> binomial(n, 2*k+1-irem(n, 2)):
seq(seq(T(n, k), k=0..ceil((n-2)/2)), n=1..20);  # Alois P. Heinz, Feb 07 2014

Flatten[ Table[ If[ OddQ[n - k], Binomial[n, k], {}], {n, 0, 15}, {k, 0, n}]] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Aug 30 2005

Corrected offset by Alois P. Heinz, Feb 07 2014

A220673 Coefficients of formal series in powers of (tan(x))^2 for tan(5*x)/tan(x).

Original entry on oeis.org

5, 40, 376, 3560, 33720, 319400, 3025400, 28657000, 271443000, 2571145000, 24354235000, 230686625000, 2185095075000, 20697517625000, 196049700875000, 1857009420625000, 17589845701875000, 166613409915625000, 1578184870646875000

Offset: 0

Wolfdieter Lang, Jan 16 2013

Formally Sum_{n>=0} a(n)*(tan(x))^(2*n) = tan(5*x)/ tan(x).
Convergence holds for x from the two open intervals (1-sqrt(6/5), 1-2/sqrt(5)) and (1+2/sqrt(5), 1+sqrt(6/5)), namely (-0.095445115, 0.105572809) and (1.894427191, 2.095445115) (10 digits).
These intervals follow from the denominator of the o.g.f. G(5,x) = (5 - 10*x + x^2)/(1 - 10*x + 5*x^2).
If one replaces x by (tan(x))^2 in this o.g.f. one obtains the formula for tan(5*x)/tan(x) in terms of (tan(x))^2. This formula is the special (n=5) solution of a general recurrence derivable from the addition theorem for tan(n*x) = tan(x + (n-1)*x), namely, with Q(n,x) := tan(n,x)/tan(x), Q(n,x) = (1 + Q(n-1,x))/(1 - v*Q(n-1,x)), where v = v(x) =(tan(x))^2, and the input is Q(1,x) = 1. Read as function of v the solution for Q(5,x) is just G(5,v) with replaced v=v(x).
See the irregular triangles A034867 and A034839 whose row polynomials N(n,x) and D(n,x), respectively, give for n >= 1 the solution to the recurrences N(n,x) = D(n-1,x) + N(n-1,x), D(n,x) = D(n-1,x) + x*N(n-1,x), with inputs N(1,x) = 1 and D(1,x) = 1. The proof by the Pascal triangle A007318 recurrence is trivial. Therefore, Q(n,x) from the preceding comment is given by Q(n,x) = N(n,-v)/D(n,-v) with v=v(x) = (tan(x))^2.
One also has, with Chebyshev's S polynomials (see A049310) Q(n,x) = tan(n*x)/tan(x) = (S(n,y) + S(n-2,y))/(S(n,y) - S(n-2,y)) = y*S(n-1,y)/(S(n,y) - S(n-2,y)) = 1/(1 - (2/y)*S(n-2,y)/S(n-1,y)), where y = y(x) = 2/sqrt(1 + (tan(x))^2). This derives from sin(n*x)/cos(n*x) in terms of Chebyshev S polynomials with argument 2*cos(x) = y(x). Note that S(n-2,y)/S(n-1,y) has the continued fraction representation 1/(y-1/(y- ... -1/(y )..(n-1)brackets..), i.e. (n-1) y's.
These calculations have been motivated by e-mails from Thomas Olsen.

			Q(5,x=0.1) = tan(0.5)/tan(0.1) = 5.444802663 (Maple 10 digits);
G(5,tan(0.1)^2) = 5.444802664;
Sum_{n>=0} a(n)*(tan(0.1))^(2*n) = 5.444802664.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. 2*A000012 (case n=2), A080923(n+1), n>=0 (case n=3), A077445(n+1), n>=0 (case n=4), A034867, A034839.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((5-10*x+x^2)/(1-10*x+5*x^2))); // G. C. Greubel, Mar 06 2018

CoefficientList[Series[(5-10*x+x^2)/(1-10*x+5*x^2), {x,0,50}], x] (* G. C. Greubel, Mar 06 2018 *)

my(x='x+O('x^30)); Vec((5-10*x+x^2)/(1-10*x+5*x^2)) \\ G. C. Greubel, Mar 06 2018

O.g.f.: G(5,x) = (5 - 10*x + x^2)/(1 - 10*x + 5*x^2).
a(n) = delta(n,0)/5 - 8*b(n) + 24*b(n+1)/5, n>=0, with Kronecker's delta and b(n):= A190987(n).
E.g.f.: (1 + 8*exp(5*x)*(3*cosh(2*sqrt(5)*x) + sqrt(5)*sinh(2*sqrt(5)*x)))/5. - Stefano Spezia, May 23 2025

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

A235803 Rectangular array read by upward antidiagonals: A(n,k) = 1 + sqrt(k)*((1+sqrt(k))^n - (1-sqrt(k))^n)/2, n,k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 5, 4, 1, 1, 9, 11, 7, 5, 1, 1, 17, 25, 19, 9, 6, 1, 1, 33, 59, 49, 29, 11, 7, 1, 1, 65, 141, 133, 81, 41, 13, 8, 1, 1, 129, 339, 361, 245, 121, 55, 15, 9, 1, 1, 257, 817, 985, 729, 401, 169, 71, 17, 10, 1

Offset: 0

L. Edson Jeffery, Jan 15 2014

			Array begins:
1,   1,    1,    1,     1,     1,     1,      1,      1,      1, ...
1,   2,    3,    4,     5,     6,     7,      8,      9,     10, ...
1,   3,    5,    7,     9,    11,    13,     15,     17,     19, ...
1,   5,   11,   19,    29,    41,    55,     71,     89,    109, ...
1,   9,   25,   49,    81,   121,   169,    225,    289,    361, ...
1,  17,   59,  133,   245,   401,   607,    869,   1193,   1585, ...
1,  33,  141,  361,   729,  1281,  2053,   3081,   4401,   6049, ...
1,  65,  339,  985,  2189,  4161,  7135,  11369,  17145,  24769, ...
1, 129,  817, 2689,  6561, 13441, 24529,  41217,  65089,  97921, ...
1, 257, 1971, 7345, 19685, 43521, 84727, 150641, 250185, 393985, ...
As a triangle:
1;
1,  1;
1,  2,  1;
1,  3,  3,  1;
1,  5,  5,  4,  1;
1,  9, 11,  7,  5,  1;
1, 17, 25, 19,  9,  6, 1;
1, 33, 59, 49, 29, 11, 7, 1; ...

Cf. A094373 (column k=1)

A(0,k) = 1, A(n,k) = 1 + k*(sum_{j=0..floor((n-1)/2)} A034867(n,j)*k^j), n>0.

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A109447 Binomial coefficients C(n,k) with n-k odd, read by rows.

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A220673 Coefficients of formal series in powers of (tan(x))^2 for tan(5*x)/tan(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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A235803 Rectangular array read by upward antidiagonals: A(n,k) = 1 + sqrt(k)*((1+sqrt(k))^n - (1-sqrt(k))^n)/2, n,k >= 0.

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