A010981 -id:A010981

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A258993

Triangle read by rows: T(n,k) = binomial(n+k,n-k), k = 0..n-1.

+10
10

1, 1, 3, 1, 6, 5, 1, 10, 15, 7, 1, 15, 35, 28, 9, 1, 21, 70, 84, 45, 11, 1, 28, 126, 210, 165, 66, 13, 1, 36, 210, 462, 495, 286, 91, 15, 1, 45, 330, 924, 1287, 1001, 455, 120, 17, 1, 55, 495, 1716, 3003, 3003, 1820, 680, 153, 19, 1, 66, 715, 3003, 6435, 8008, 6188, 3060, 969, 190, 21

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

OFFSET

1,3

COMMENTS

T(n,k) = A085478(n,k) = A007318(A094727(n),A004736(k)), k = 0..n-1;

rounded(T(n,k)/(2*k+1)) = A258708(n,k);

rounded(sum(T(n,k)/(2*k+1)): k = 0..n-1) = A000967(n).

LINKS

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

FORMULA

T(n,k) = A085478(n,k) = A007318(A094727(n),A004736(k)), k = 0..n-1;

rounded(T(n,k)/(2*k+1)) = A258708(n,k);

rounded(sum(T(n,k)/(2*k+1)): k = 0..n-1) = A000967(n).

EXAMPLE

. n\k | 0 1 2 3 4 5 6 7 8 9 10 11

. -----+-----------------------------------------------------------

. 1 | 1

. 2 | 1 3

. 3 | 1 6 5

. 4 | 1 10 15 7

. 5 | 1 15 35 28 9

. 6 | 1 21 70 84 45 11

. 7 | 1 28 126 210 165 66 13

. 8 | 1 36 210 462 495 286 91 15

. 9 | 1 45 330 924 1287 1001 455 120 17

. 10 | 1 55 495 1716 3003 3003 1820 680 153 19

. 11 | 1 66 715 3003 6435 8008 6188 3060 969 190 21

. 12 | 1 78 1001 5005 12870 19448 18564 11628 4845 1330 231 23 .

MATHEMATICA

Table[Binomial[n+k, n-k], {n, 1, 12}, {k, 0, n-1}]//Flatten (* G. C. Greubel, Aug 01 2019 *)

PROG

(Haskell)

a258993 n k = a258993_tabl !! (n-1) !! k

a258993_row n = a258993_tabl !! (n-1)

a258993_tabl = zipWith (zipWith a007318) a094727_tabl a004736_tabl

(PARI) T(n, k) = binomial(n+k, n-k);

for(n=1, 12, for(k=0, n-1, print1(T(n, k), ", "))) \\ G. C. Greubel, Aug 01 2019

(Magma) [Binomial(n+k, n-k): k in [0..n-1], n in [1..12]]; // G. C. Greubel, Aug 01 2019

(Sage) [[binomial(n+k, n-k) for k in (0..n-1)] for n in (1..12)] # G. C. Greubel, Aug 01 2019

(GAP) Flat(List([1..12], n-> List([0..n-1], k-> Binomial(n+k, n-k) ))); # G. C. Greubel, Aug 01 2019

CROSSREFS

If a diagonal of 1's is added on the right, this becomes A085478.

Essentially the same as A143858.

Cf. A007318, A004736, A094727.

Cf. A027941 (row sums), A117671 (central terms), A143858, A000967, A258708.

T(n,k): A000217 (k=1), A000332 (k=2), A000579 (k=3), A000581 (k=4), A001287 (k=5), A010965 (k=6), A010967 (k=7), A010969 (k=8), A010971 (k=9), A010973 (k=10), A010975 (k=11), A010977 (k=12), A010979 (k=13), A010981 (k=14), A010983 (k=15), A010985 (k=16), A010987 (k=17), A010989 (k=18), A010991 (k=19), A010993 (k=20), A010995 (k=21), A010997 (k=22), A010999 (k=23), A011001 (k=24), A017714 (k=25), A017716 (k=26), A017718 (k=27), A017720 (k=28), A017722 (k=29), A017724 (k=30), A017726 (k=31), A017728 (k=32), A017730 (k=33), A017732 (k=34), A017734 (k=35), A017736 (k=36), A017738 (k=37), A017740 (k=38), A017742 (k=39), A017744 (k=40), A017746 (k=41), A017748 (k=42), A017750 (k=43), A017752 (k=44), A017754 (k=45), A017756 (k=46), A017758 (k=47), A017760 (k=48), A017762 (k=49), A017764 (k=50).

T(n+k,n): A005408 (k=1), A000384 (k=2), A000447 (k=3), A053134 (k=4), A002299 (k=5), A053135 (k=6), A053136 (k=7), A053137 (k=8), A053138 (k=9), A196789 (k=10).

Cf. A165253.

KEYWORD

nonn,tabl

AUTHOR

Reinhard Zumkeller, Jun 22 2015

STATUS

approved

A010984

Binomial coefficient C(n,31).

+10
6

1, 32, 528, 5984, 52360, 376992, 2324784, 12620256, 61523748, 273438880, 1121099408, 4280561376, 15338678264, 51915526432, 166871334960, 511738760544, 1503232609098, 4244421484512, 11554258485616, 30405943383200, 77535155627160, 191991813933920

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

OFFSET

31,2

LINKS

T. D. Noe, Table of n, a(n) for n = 31..1000

Index entries for linear recurrences with constant coefficients, signature (32, -496, 4960, -35960, 201376, -906192, 3365856, -10518300, 28048800, -64512240, 129024480, -225792840, 347373600, -471435600, 565722720, -601080390, 565722720, -471435600, 347373600, -225792840, 129024480, -64512240, 28048800, -10518300, 3365856, -906192, 201376, -35960, 4960, -496, 32, -1).

FORMULA

G.f.: x^31/(1-x)^32. - Zerinvary Lajos, Dec 19 2008; adapted to offset by Enxhell Luzhnica, Jan 21 2017

From Amiram Eldar, Dec 12 2020: (Start)

Sum_{n>=31} 1/a(n) = 31/30.

Sum_{n>=31} (-1)^(n+1)/a(n) = A001787(31)*log(2) - A242091(31)/30! = 33285996544*log(2) - 6717121856795533085173/291136195350 = 0.9704936372... (End)

MAPLE

seq(binomial(n, 31), n=31..53); # Zerinvary Lajos, Dec 19 2008

MATHEMATICA

Table[Binomial[n, 31], {n, 31, 60}] (* Vladimir Joseph Stephan Orlovsky, Apr 26 2011 *)

PROG

(Magma) [Binomial(n, 31): n in [31..70]]; // Vincenzo Librandi, Jun 12 2013

(PARI) x='x+O('x^50); Vec(x^31/(1-x)^32) \\ G. C. Greubel, Nov 23 2017

CROSSREFS

Cf. A010980, A010981, A010982, A001787, A242091

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A010982

Binomial coefficient C(n,29).

+10
5

1, 30, 465, 4960, 40920, 278256, 1623160, 8347680, 38608020, 163011640, 635745396, 2311801440, 7898654920, 25518731280, 78378960360, 229911617056, 646626422970, 1749695026860, 4568648125690, 11541847896480, 28277527346376, 67327446062800, 156077261327400

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

OFFSET

29,2

LINKS

T. D. Noe, Table of n, a(n) for n = 29..1000

Index entries for linear recurrences with constant coefficients, signature (30, -435, 4060, -27405, 142506, -593775, 2035800, -5852925, 14307150, -30045015, 54627300, -86493225, 119759850, -145422675, 155117520, -145422675, 119759850, -86493225, 54627300, -30045015, 14307150, -5852925, 2035800, -593775, 142506, -27405, 4060, -435, 30, -1).

FORMULA

G.f.: x^29/(1-x)^30. - Zerinvary Lajos, Dec 19 2008; adapted to offset by Enxhell Luzhnica, Jan 21 2017

From Amiram Eldar, Dec 12 2020: (Start)

Sum_{n>=29} 1/a(n) = 29/28.

Sum_{n>=29} (-1)^(n+1)/a(n) = A001787(29)*log(2) - A242091(29)/28! = 7784628224*log(2) - 108340675094713923269/20078358300 = 0.9686369528... (End)

MAPLE

seq(binomial(n, 29), n=29..53); # Zerinvary Lajos, Dec 19 2008

MATHEMATICA

Table[Binomial[n, 29], {n, 29, 60}] (* Vladimir Joseph Stephan Orlovsky, Apr 26 2011 *)

PROG

(Magma) [Binomial(n, 29): n in [29..60]]; // Vincenzo Librandi, Jun 12 2013

(PARI) x='x+O('x^50); Vec(x^29/(1-x)^30) \\ G. C. Greubel, Nov 23 2017

CROSSREFS

Cf. A010980, A010981, A001787, A242091.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A096754

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

+10
1

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

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

OFFSET

1,7

COMMENTS

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

LINKS

Table of n, a(n) for n=1..89.

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

EXAMPLE

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 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

MATHEMATICA

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

CROSSREFS

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.

KEYWORD

sign,tabl

AUTHOR

Robert G. Wilson v, Jul 07 2004

STATUS

approved

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