A017715 -id:A017715

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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

A017716

Binomial coefficients C(n,52).

+10
5

1, 53, 1431, 26235, 367290, 4187106, 40475358, 341149446, 2558620845, 17341763505, 107518933731, 615790256823, 3284214703056, 16421073515280, 77413632286320, 345780890878896, 1469568786235308, 5964720367660956

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

OFFSET

52,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 52..10000

FORMULA

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

G.f.: x^52/(1-x)^53.

E.g.f.: x^52*exp(x)/52!. (End)

From Amiram Eldar, Dec 16 2020: (Start)

Sum_{n>=52} 1/a(n) = 52/51.

Sum_{n>=52} (-1)^n/a(n) = A001787(52)*log(2) - A242091(52)/51! = 117093590311632896*log(2) - 120926939503504532846299231985163098 / 1489925242425959955 = 0.9817952764... (End)

MATHEMATICA

Table[Binomial[n, 52], {n, 52, 80}] (* Vladimir Joseph Stephan Orlovsky, May 16 2011 *)

PROG

(Sage) [binomial(n, 52) for n in range(52, 70)] # Zerinvary Lajos, May 23 2009

(PARI) for(n=52, 80, print1(binomial(n, 52), ", ")) \\ G. C. Greubel, Nov 03 2018

(Magma) [Binomial(n, 52): n in [52..80]]; // G. C. Greubel, Nov 03 2018

CROSSREFS

Cf. A017713, A017715, A001787, A242091.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

A017717

Binomial coefficients C(n,53).

+10
4

1, 54, 1485, 27720, 395010, 4582116, 45057474, 386206920, 2944827765, 20286591270, 127805525001, 743595781824, 4027810484880, 20448884000160, 97862516286480, 443643407165376, 1913212193400684, 7877932561061640

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

OFFSET

53,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 53..10000

FORMULA

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

G.f.: x^53/(1-x)^54.

E.g.f.: x^53*exp(x)/53!. (End)

From Amiram Eldar, Dec 16 2020: (Start)

Sum_{n>=53} 1/a(n) = 53/52.

Sum_{n>=53} (-1)^(n+1)/a(n) = A001787(53)*log(2) - A242091(53)/52! = 238690780250636288*log(2) - 12818255587371480560673756439003166003 / 77476112606149917660 = 0.9821211403... (End)

MATHEMATICA

Table[Binomial[n, 53], {n, 53, 80}] (* Vladimir Joseph Stephan Orlovsky, May 16 2011 *)

PROG

(Sage) [binomial(n, 53) for n in range(53, 71)] # Zerinvary Lajos, May 23 2009

(PARI) for(n=53, 80, print1(binomial(n, 53), ", ")) \\ G. C. Greubel, Nov 03 2018

(Magma) [Binomial(n, 53): n in [53..80]]; // G. C. Greubel, Nov 03 2018

CROSSREFS

Cf. A017713, A017715, A017716, A001787, A242091.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

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

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