A010971 -id:A010971 - cp's OEIS frontend

A258993 Triangle read by rows: T(n,k) = binomial(n+k,n-k), k = 0..n-1.

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

Offset: 1

Reinhard Zumkeller, Jun 22 2015

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).

			.  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  .

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

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.

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

a258993 n k = a258993_tabl !! (n-1) !! k
a258993_row n = a258993_tabl !! (n-1)
a258993_tabl = zipWith (zipWith a007318) a094727_tabl a004736_tabl

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

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

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

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

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).

A010976 Binomial coefficient C(n,23).

Original entry on oeis.org

1, 24, 300, 2600, 17550, 98280, 475020, 2035800, 7888725, 28048800, 92561040, 286097760, 834451800, 2310789600, 6107086800, 15471286560, 37711260990, 88732378800, 202112640600, 446775310800, 960566918220, 2012616400080, 4116715363800, 8233430727600

Offset: 23

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 23..1000
Index entries for linear recurrences with constant coefficients, signature (24,-276,2024,-10626,42504,-134596,346104,-735471,1307504,-1961256,2496144,-2704156,2496144,-1961256,1307504,-735471,346104,-134596,42504,-10626,2024,-276,24,-1).

Pascal's triangle A007318. [Zerinvary Lajos, Aug 04 2008]

Cf. A010970, A010971, A010972, A001787, A242091.

[Binomial(n,23): n in [23..90]]; // Vincenzo Librandi, Mar 26 2011

seq(binomial(n,23), n=23..43); # Zerinvary Lajos, Aug 04 2008

Table[Binomial[n,23],{n,23,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 26 2011 *)

for(n=23, 50, print1(binomial(n,23), ", ")) \\ G. C. Greubel, Nov 23 2017

a(n) = n/(n-23) * a(n-1) for n > 23. - Vincenzo Librandi, Mar 26 2011
G.f.: x^23/(1-x)^24. - G. C. Greubel, Nov 23 2017
From Amiram Eldar, Dec 11 2020: (Start)
Sum_{n>=23} 1/a(n) = 23/22.
Sum_{n>=23} (-1)^(n+1)/a(n) = A001787(23)*log(2) - A242091(23)/22! = 96468992*log(2) - 1945773591174209/29099070 = 0.9613305695... (End)

A010977 a(n) = binomial coefficient C(n,24).

Original entry on oeis.org

1, 25, 325, 2925, 20475, 118755, 593775, 2629575, 10518300, 38567100, 131128140, 417225900, 1251677700, 3562467300, 9669554100, 25140840660, 62852101650, 151584480450, 353697121050, 800472431850, 1761039350070, 3773655750150, 7890371113950, 16123801841550

Offset: 24

N. J. A. Sloane

nonn

T. D. Noe, Table of n, a(n) for n = 24..1000
Index entries for linear recurrences with constant coefficients, signature (25, -300, 2300, -12650, 53130, -177100, 480700, -1081575, 2042975, -3268760, 4457400, -5200300, 5200300, -4457400, 3268760, -2042975, 1081575, -480700, 177100, -53130, 12650, -2300, 300, -25, 1).

Pascal's triangle A007318 diagonal. - Zerinvary Lajos, Aug 04 2008

Cf. A010970, A010971, A010972, A001787, A242091.

[ Binomial(n,24): n in [24..90]]; // Vincenzo Librandi, Mar 26 2011

seq(binomial(n,24),n=24..41); # Zerinvary Lajos, Aug 04 2008

Table[Binomial[n,24],{n,24,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 26 2011 *)

x='x+O('x^50); Vec(x^24/(1-x)^25) \\ G. C. Greubel, Nov 23 2017

G.f.: x^24/(1-x)^25. - Zerinvary Lajos, Aug 04 2008 [Corrected by Georg Fischer, May 19 2019]
a(n) = n/(n-24) * a(n-1), n > 24. - Vincenzo Librandi, Mar 26 2011
From Amiram Eldar, Dec 11 2020: (Start)
Sum_{n>=24} 1/a(n) = 24/23.
Sum_{n>=24} (-1)^n/a(n) = A001787(24)*log(2) - A242091(24)/23! = 201326592*log(2) - 15566188845789952/111546435 = 0.9627768409... (End)

A010978 a(n) = binomial(n,25).

Original entry on oeis.org

1, 26, 351, 3276, 23751, 142506, 736281, 3365856, 13884156, 52451256, 183579396, 600805296, 1852482996, 5414950296, 15084504396, 40225345056, 103077446706, 254661927156, 608359048206, 1408831480056, 3169870830126, 6943526580276, 14833897694226, 30957699535776

Offset: 25

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 25..1000
Index entries for linear recurrences with constant coefficients, signature (26, -325, 2600, -14950, 65780, -230230, 657800, -1562275, 3124550, -5311735, 7726160, -9657700, 10400600, -9657700, 7726160, -5311735, 3124550, -1562275, 657800, -230230, 65780, -14950, 2600, -325, 26, -1).

Cf. A010970, A010971, A010972, A001787, A242091.

[Binomial(n, 25): n in [25..50]]; // Vincenzo Librandi, Jun 12 2013

seq(binomial(n,25),n=25..41); # Zerinvary Lajos, Aug 18 2008

Table[Binomial[n,25],{n,25,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 26 2011 *)

x='x+O('x^50); Vec(x^25/(1-x)^26) \\ G. C. Greubel, Nov 23 2017

From Zerinvary Lajos, Aug 18 2008: (Start)
a(n) = C(n,25), n >= 25.
G.f.: x^25/(1-x)^26. (End) [G.f. corrected by Georg Fischer, May 19 2019]
From Amiram Eldar, Dec 11 2020: (Start)
Sum_{n>=25} 1/a(n) = 25/24.
Sum_{n>=25} (-1)^(n+1)/a(n) = A001787(25)*log(2) - A242091(25)/24! = 419430400*log(2) - 155661889015631695/535422888 = 0.9641184185... (End)

A010979 Binomial coefficient C(n,26).

Original entry on oeis.org

1, 27, 378, 3654, 27405, 169911, 906192, 4272048, 18156204, 70607460, 254186856, 854992152, 2707475148, 8122425444, 23206929840, 63432274896, 166509721602, 421171648758, 1029530696964, 2438362177020, 5608233007146, 12551759587422, 27385657281648

Offset: 26

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 26..1000
Index entries for linear recurrences with constant coefficients, signature (27, -351, 2925, -17550, 80730, -296010, 888030, -2220075, 4686825, -8436285, 13037895, -17383860, 20058300, -20058300, 17383860, -13037895, 8436285, -4686825, 2220075, -888030, 296010, -80730, 17550, -2925, 351, -27, 1).

Cf. A010970, A010971, A010972, A001787, A242091.

[Binomial(n, 26): n in [26..60]]; // Vincenzo Librandi, Jun 12 2013

seq(binomial(n,26),n=26..41); # Zerinvary Lajos, Aug 18 2008

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

x='x+O('x^50); Vec(x^26/(1-x)^27) \\ G. C. Greubel, Nov 23 2017

G.f.: x^26/(1-x)^27. - Zerinvary Lajos, Aug 18 2008; adapted to offset by Enxhell Luzhnica, Jan 21 2017
From Amiram Eldar, Dec 11 2020: (Start)
Sum_{n>=26} 1/a(n) = 26/25.
Sum_{n>=26} (-1)^n/a(n) = A001787(26)*log(2) - A242091(26)/25! = 872415232*log(2) - 155661889283343139/257414850 = 0.9653663105... (End)

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}]]

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A258993 Triangle read by rows: T(n,k) = binomial(n+k,n-k), k = 0..n-1.

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A010976 Binomial coefficient C(n,23).

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A010977 a(n) = binomial coefficient C(n,24).

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A010978 a(n) = binomial(n,25).

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A010979 Binomial coefficient C(n,26).

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

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