A017717 -id:A017717

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A238801

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

+10
8

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Row sums are powers of 2.

LINKS

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

FORMULA

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.

EXAMPLE

Triangle begins:

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.

MATHEMATICA

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

PROG

(PARI) 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

CROSSREFS

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.

KEYWORD

nonn,tabl

AUTHOR

Philippe Deléham, Mar 05 2014

STATUS

approved

A095704

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

+10
4

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, 13, 0, -286, 0, 1287, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..84.

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

FORMULA

T(n,k) = C(n+1,k+1)*sin(Pi*(k+1)/2). - Paul Barry, May 21 2006

EXAMPLE

The trigonometric expansion of sin(4x) is 4*cos(x)^3*sin(x) - 4*cos(x)*sin(x)^3, so the fourth row is 4, 0, -4, 0.

Triangle begins:

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

MATHEMATICA

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

CROSSREFS

First column is A000027 = C(n, 1), third column is A000292 = C(n, 3), fifth column is A000389 = C(n, 5), seventh column is A000580 = C(n, 7), ninth column is A000582 = C(n, 9).

A001288 = C(n, 11), A010966 = C(n, 13), A010968 = C(n, 15), A010970 = C(n, 17), A010972 = C(n, 19),

A010974 = C(n, 21), A010976 = C(n, 23), A010978 = C(n, 25), A010980 = C(n, 27), A010982 = C(n, 29),

A010984 = C(n, 31), A010986 = C(n, 33), A010988 = C(n, 35), A010990 = C(n, 37), A010992 = C(n, 39),

A010994 = C(n, 41), A010996 = C(n, 43), A010998 = C(n, 45), A011000 = C(n, 47), A017713 = C(n, 49)

Another version of the triangle in A034867. Cf. A096754.

A017715 = C(n, 51), A017717 = C(n, 53), A017719 = C(n, 55), A017721 = C(n, 57), etc.

KEYWORD

sign,tabl

AUTHOR

Robert G. Wilson v, Jul 06 2004

STATUS

approved

A017720

Binomial coefficients C(n,56).

+10
3

1, 57, 1653, 32509, 487635, 5949147, 61474519, 553270671, 4426165368, 31966749880, 210980549208, 1285063345176, 7282025622664, 38650751381832, 193253756909160, 914734449370024, 4116305022165108, 17675898036356052

(list; graph; refs; listen; history; text; internal format)

OFFSET

56,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 56..10000

FORMULA

From G. C. Greubel, Nov 03 2018: (Start)

G.f.: x^56/(1-x)^57.

E.g.f.: x^56*exp(x)/56!. (End)

From Amiram Eldar, Dec 16 2020: (Start)

Sum_{n>=56} 1/a(n) = 56/55.

Sum_{n>=56} (-1)^n/a(n) = A001787(56)*log(2) - A242091(56)/55! = 2017612633061982208*log(2) - 41018417879588738008814416366926778496 / 29330242629471040257 = 0.9830322375... (End)

MATHEMATICA

Table[Binomial[n, 56], {n, 56, 80}] (* Vladimir Joseph Stephan Orlovsky, May 16 2011 *)

PROG

(Sage) [binomial(n, 56) for n in range(56, 74)] # Zerinvary Lajos, May 23 2009

(PARI) for(n=56, 80, print1(binomial(n, 56), ", ")) \\ G. C. Greubel, Nov 03 2018

(Magma) [Binomial(n, 56): n in [56..80]]; // G. C. Greubel, Nov 03 2018

CROSSREFS

Cf. A017717, A017719, A001787, A242091.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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