A010967 -id:A010967 - cp's OEIS frontend

A110555 Triangle of partial sums of alternating binomial coefficients: T(n, k) = Sum_{j=0..k} binomial(n, j)*(-1)^j, for n >= 0, 0 <= k <= n.

1, 1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -3, 3, -1, 0, 1, -4, 6, -4, 1, 0, 1, -5, 10, -10, 5, -1, 0, 1, -6, 15, -20, 15, -6, 1, 0, 1, -7, 21, -35, 35, -21, 7, -1, 0, 1, -8, 28, -56, 70, -56, 28, -8, 1, 0, 1, -9, 36, -84, 126, -126, 84, -36, 9, -1, 0, 1, -10, 45, -120, 210, -252, 210, -120

Offset: 0

Reinhard Zumkeller, Jul 27 2005

			Triangle T(n, k) starts:
  [0] 1;
  [1] 1,  0;
  [2] 1, -1,  0;
  [3] 1, -2,  1,   0;
  [4] 1, -3,  3,  -1,  0;
  [5] 1, -4,  6,  -4,  1,   0;
  [6] 1, -5, 10, -10,  5,  -1,  0;
  [7] 1, -6, 15, -20, 15,  -6,  1,  0;
  [8] 1, -7, 21, -35, 35, -21,  7, -1,  0.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Ângela Mestre and José Agapito, A Family of Riordan Group Automorphisms, J. Int. Seq., Vol. 22 (2019), Article 19.8.5.
Index entries for triangles and arrays related to Pascal's triangle

T(n,1)  = -n + 1 for n>0;
T(n,2)  =  A000217(n-2) for n > 1;
T(n,3)  = -A000292(n-4) for n > 2;
T(n,4)  =  A000332(n-1) for n > 3;
T(n,5)  = -A000389(n-1) for n > 5;
T(n,6)  =  A000579(n-1) for n > 6;
T(n,7)  = -A000580(n-1) for n > 7;
T(n,8)  =  A000581(n-1) for n > 8;
T(n,9)  = -A000582(n-1) for n > 9;
T(n,10) =  A001287(n-1) for n > 10;
T(n,11) = -A001288(n-1) for n > 11;
T(n,12) =  A010965(n-1) for n > 12;
T(n,13) = -A010966(n-1) for n > 13;
T(n,14) =  A010967(n-1) for n > 14;
T(n,15) = -A010968(n-1) for n > 15;
T(n,16) =  A010969(n-1) for n > 16.
Cf. A071919 (variant), A000007 (row sums), A110556 (central terms).
Cf. A008949, A007318.

T := (n, k) -> (-1)^k * binomial(n-1, k):
seq(print(seq(T(n, k), k = 0..n)), n = 0..7); # Peter Luschny, Apr 13 2023

T[0, 0] := 1;  T[n_, n_] := 0; T[n_, k_] := (-1)^k*Binomial[n - 1, k]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 31 2017 *)

concat(1, for(n=1,10, for(k=0,n, print1(if(k != n, (-1)^k*binomial(n-1,k), 0), ", ")))) \\ G. C. Greubel, Aug 31 2017

T(n, 0) = 1, T(n, n) = 0^n, T(n, k) = -T(n-1, k-1) + T(n-1, k), for 0 < k < n.
T(n, k) = binomial(n-1, k)*(-1)^k, 0 <= k < n, T(n, n) = 0^n.
T(n, n-k-1) = -T(n, k), for 0 < k < n.
T(n, k) = A071919(n, k)*(-1)^k and A071919(n, k) = abs(T(n, k)).
Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [0, -1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 05 2005
G.f.: (1 + x*y) / (1 + x*y - x). - R. J. Mathar, Aug 11 2015

Typo in name corrected by Andrey Zabolotskiy, Feb 22 2022

Offset corrected by Peter Luschny, Apr 13 2023

A010965 a(n) = binomial(n,12).

Original entry on oeis.org

1, 13, 91, 455, 1820, 6188, 18564, 50388, 125970, 293930, 646646, 1352078, 2704156, 5200300, 9657700, 17383860, 30421755, 51895935, 86493225, 141120525, 225792840, 354817320, 548354040, 834451800, 1251677700, 1852482996, 2707475148, 3910797436, 5586853480

Offset: 12

N. J. A. Sloane

nonn

Coordination sequence for 12-dimensional cyclotomic lattice Z[zeta_13].

In this sequence only 13 is prime. - Artur Jasinski, Dec 02 2007

T. D. Noe, Table of n, a(n) for n = 12..1000
Matthias Beck and Serkan Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
Index entries for linear recurrences with constant coefficients, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).

Cf. A000581, A010966, A010967, A001787, A242091.

[Binomial(n, 12): n in [12..100]]; // Vincenzo Librandi, Apr 22 2011

seq(binomial(n,12),n=12..36); # Zerinvary Lajos, Aug 06 2008

Table[Binomial[n,12],{n,12,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)

for(n=12, 50, print1(binomial(n,12), ", ")) \\ G. C. Greubel, Aug 31 2017

a(n) = A110555(n+1,12). - Reinhard Zumkeller, Jul 27 2005
a(n+11) = n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)(n+9)(n+10)(n+11)/12!. - Artur Jasinski, Dec 02 2007, R. J. Mathar, Jul 07 2009
G.f.: x^12/(1-x)^13. - Zerinvary Lajos, Aug 06 2008, R. J. Mathar, Jul 07 2009
From Amiram Eldar, Dec 10 2020: (Start)
Sum_{n>=12} 1/a(n) = 12/11.
Sum_{n>=12} (-1)^n/a(n) = A001787(12)*log(2) - A242091(12)/11! = 24576*log(2) - 3934820/231 = 0.9322955884... (End)

Some formulas referring to other offsets corrected by R. J. Mathar, Jul 07 2009

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

Original entry on oeis.org

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).

A155856 Triangle T(n,k) = binomial(2n-k, k)(n-k)!, read by rows.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 10, 6, 1, 24, 42, 30, 10, 1, 120, 216, 168, 70, 15, 1, 720, 1320, 1080, 504, 140, 21, 1, 5040, 9360, 7920, 3960, 1260, 252, 28, 1, 40320, 75600, 65520, 34320, 11880, 2772, 420, 36, 1, 362880, 685440, 604800, 327600, 120120, 30888, 5544, 660, 45, 1

Offset: 0

Paul Barry, Jan 29 2009

Row sums of B^{-1}*A155856*B^{-1} are A000166 with B=A007318.

Downward diagonals T(n+j, n) = j!*binomial(n+j, n) = j!*seq(j), where seq(j) are sequences A010965, A010967, ..., A011101, A017714, A017716, ..., A017764, for 6 <= j <= 50, respectively. - G. C. Greubel, Jun 04 2021

			Triangle begins:
     1;
     1,    1;
     2,    3,    1;
     6,   10,    6,    1;
    24,   42,   30,   10,    1;
   120,  216,  168,   70,   15,   1;
   720, 1320, 1080,  504,  140,  21,  1;
  5040, 9360, 7920, 3960, 1260, 252, 28, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 58.

Cf. A155857 (row sums), A155858 (diagonal sums).

Table[Binomial[2n-k,k](n-k)!,{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Mar 24 2017 *)

flatten([[factorial(n-k)*binomial(2*n-k, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 04 2021

T(n,k) = binomial(2*n-k, k)*(n-k)!.
Sum_{k=0..n} T(n, k) = A155857(n)
Sum_{k=0..floor(n/2)} T(n-k, k) = A155858(n) (diagonal sums).
G.f.: 1/(1-xy-x/(1-xy-x/(1-xy-2x/(1-xy-2x/(1-xy-3x/(1-.... (continued fraction).
From G. C. Greubel, Jun 04 2021: (Start)
  T(n, 0) = A000142(n).        T(n+1, n) = A000217(n+1).
T(n+1, 1) = A007680(n).        T(n+2, n) = A034827(n+4).
T(n+2, 2) = A175925(n).        T(n+3, n) = A253946(n).
T(2*n, n) = A064352(n)         T(n+4, n) = 4!*A000581(n).
T(n+1, n) = A000217(n+1).      T(n+5, n) = 5!*A001287(n).  (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}]]

A206294 Riordan array (1, x/(1-x)^3).

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 6, 6, 1, 0, 10, 21, 9, 1, 0, 15, 56, 45, 12, 1, 0, 21, 126, 165, 78, 15, 1, 0, 28, 252, 495, 364, 120, 18, 1, 0, 36, 462, 1287, 1365, 680, 171, 21, 1, 0, 45, 792, 3003, 4368, 3060, 1140, 231, 24, 1

Offset: 0

Philippe Deléham, Feb 05 2012

The convolution triangle of the triangular numbers A000217. - Peter Luschny, Oct 07 2022

			Triangle begins:
1
0, 1
0, 3, 1
0, 6, 6, 1
0, 10, 21, 9, 1
0, 15, 56, 45, 12, 1
0, 21, 126, 165, 78, 15, 1
0, 28, 252, 495, 364, 120, 18, 1
0, 36, 462, 1287, 1365, 680, 171, 21, 1
0, 45, 792, 3003, 4368, 3060, 1140, 231, 24, 1
0, 55, 1287, 6435, 12376, 11628, 5985, 1771, 300, 27, 1
0, 66, 2002, 12870, 31824, 38760, 26324, 10626, 2600, 378, 30, 1

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. Columns: A000007, A000217 (triangular numbers), A000389, A000581, A001288, A010967..(+3)..A011000, A017714..(+3)..A017762.
Row sums are A052529.
Cf. A127893.

# Uses function PMatrix from A357368.
PMatrix(10, n -> n * (n + 1) / 2); # Peter Luschny, Oct 07 2022

Table[If[n == 0 && k == 0 , 1, Binomial[n - 1 + 2 k, n - k]], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 25 2017 *)

{T(n,k)=polcoeff(1/(1-x+x*O(x^(n-k)))^(3*k),n-k)}

{T(n,k)=polcoeff(polcoeff((1-x)^3/((1-x)^3-y*x +x*O(x^n)),n,x),k,y)}
for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))

Triangle T(n,k), read by rows, given by (0, 3, -1, 2/3, -1/6, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
T(n,0) = 0^n, T(n,k) = C(n-1+2k, n-k) for k > 0.
T(n,n) = 1, T(k+1,k) = 3*k = A008585(k), T(k+2,k) = A081266(k).
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A052529(n), A052910(n) for x = 0, 1, 2 respectively.
G.f.: (1-x)^3/((1-x)^3-y*x).

A188225 Number of ways to select 15 knights from n knights sitting at a round table if no adjacent knights are chosen.

Original entry on oeis.org

2, 31, 256, 1496, 6936, 27132, 93024, 286824, 810084, 2124694, 5230016, 12183560, 27041560, 57500460, 117675360, 232676280, 445962870, 830905245, 1508593920, 2674776720, 4639918800, 7887861960, 13160496960, 21578373360, 34810394760

Offset: 30

Zoltán Lőrincz, Mar 30 2011

nonn

			For n = 30 a(30) = C(15, 15) + C(14, 14) = 2

a(n) = A010968(n - 15) + A010967(n - 16)

Table[Binomial[n-15,15]+Binomial[n-16,14],{n,30,60}] (* Harvey P. Dale, May 16 2018 *)

a(n) = C(n - 15, 15) + C(n - 16, 14)

cp's OEIS Frontend

A110555 Triangle of partial sums of alternating binomial coefficients: T(n, k) = Sum_{j=0..k} binomial(n, j)*(-1)^j, for n >= 0, 0 <= k <= n.

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A010965 a(n) = binomial(n,12).

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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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A155856 Triangle T(n,k) = binomial(2*n-k, k)*(n-k)!, read by rows.

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

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Mathematica

A206294 Riordan array (1, x/(1-x)^3).

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Maple

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PARI

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A155856 Triangle T(n,k) = binomial(2n-k, k)(n-k)!, read by rows.