A000346 -id:A000346 - cp's OEIS frontend

A386567 a(n) = Sum_{k=0..n-1} binomial(6k-1,k) binomial(6n-6k,n-k-1).

0, 1, 17, 268, 4129, 62955, 954392, 14417376, 217279857, 3269099590, 49125066135, 737516631908, 11064270530632, 165889863957065, 2486052264852180, 37241727274394640, 557707191712371729, 8349517132932620730, 124971965902300790390, 1870139909398530770760

Offset: 0

Seiichi Manyama, Jul 26 2025

nonn

			(1/5) * log( Sum_{k>=0} binomial(6*k-1,k)*x^k ) = x + 17*x^2/2 + 268*x^3/3 + 4129*x^4/4 + 12591*x^5 + ...

Cf. A000346, A062236, A386565, A386566.
Cf. A079679, A386368, A386615, A386616.
Cf. A002295, A130564.

a(n) = sum(k=0, n-1, binomial(6*k-1, k)*binomial(6*n-6*k, n-k-1));

my(N=20, x='x+O('x^N), g=sum(k=0, N, binomial(6*k, k)/(5*k+1)*x^k)); concat(0, Vec(g*(g-1)/(6-5*g)^2))

G.f.: g*(g-1)/(6-5*g)^2 where g=1+x*g^6.
G.f.: g/(1-6*g)^2 where g*(1-g)^5 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/5) * log( Sum_{k>=0} binomial(6*k-1,k)*x^k ).
a(n) = Sum_{k=0..n-1} binomial(6*k-1+l,k) * binomial(6*n-6*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(6*n,k).
a(n) = Sum_{k=0..n-1} 6^(n-k-1) * binomial(5*n+k,k).

A119245 Triangle, read by rows, defined by: T(n,k) = (4k+1)binomial(2n+1, n-2k)/(2n+1) for n >= 2k >= 0.

Original entry on oeis.org

1, 1, 2, 1, 5, 5, 14, 20, 1, 42, 75, 9, 132, 275, 54, 1, 429, 1001, 273, 13, 1430, 3640, 1260, 104, 1, 4862, 13260, 5508, 663, 17, 16796, 48450, 23256, 3705, 170, 1, 58786, 177650, 95931, 19019, 1309, 21, 208012, 653752, 389367, 92092, 8602, 252, 1

Offset: 0

Paul D. Hanna, May 10 2006

Closely related to triangle A118919.
Row n contains 1+floor(n/2) terms.
From Peter Bala, Mar 20 2009: (Start)
Combinatorial interpretations of T(n,k):
1) The number of standard tableaux of shape (n-2*k,n+2*k).
2) The entries in column k are (with an offset of 2*k) the number of n-th generation vertices in the tree of sequences with unit increase labeled by 4*k. See [Sunik, Theorem 4]. (End)

			Triangle begins:
     1;
     1;
     2,     1;
     5,     5;
    14,    20,    1;
    42,    75,    9;
   132,   275,   54,   1;
   429,  1001,  273,  13;
  1430,  3640, 1260, 104,  1;
  4862, 13260, 5508, 663, 17; ...

Zoran Sunic, Self describing sequences and the Catalan family tree, Elect. J. Combin., 10 (No. 1, 2003). - _Peter Bala_, Mar 20 2009

Cf. A119244 (eigenvector), A088218, A000108, A000344, A001392; A118919 (variant), A158483; A002057, A002894.

f1 = (1-Sqrt[1-4*x])/(2*x);
DeleteCases[CoefficientList[Normal@Series[f1/(1 - x^2*y*f1^4),{x,0,10},{y,0,5}],{x,y}],0,Infinity]//TableForm  (* Bradley Klee, Feb 26 2018 *)
Table[(1+4*k)/(n+1+2*k)*Binomial[2*n,n+2*k],{n,0,10},{k,0,Floor[n/2]}]//TableForm (* Bradley Klee, Feb 26 2018 *)

T(n,k)=(4*k+1)*binomial(2*n+1,n-2*k)/(2*n+1)

G.f.: A(x,y) = f/(1-x^2*y*f^4), where f=(1-sqrt(1-4*x))/(2*x) is the Catalan g.f. (A000108).
Row sums equal A088218(n) = C(2*n-1,n).
T(n,0) = A000108(n) (the Catalan numbers).
T(n,1) = A000344(n).
T(n,2) = A001392(n).
Sum_{k=0..floor(n/2)} k*T(n,k) = A000346(n-2).
Eigenvector is defined by: A119244(n) = Sum_{k=0..[n\2]} T(n,k)*A119244(k).
...
T(n,k) = (4*k+1)/(n+2*k+1)*binomial(2*n,n+2*k). Compare with A158483. - Peter Bala, Mar 20 2009
T(n,k) = A039599(n, 2*k). - Johannes W. Meijer, Sep 04 2013
A002894(n) = Sum_{k=0..floor(n/2)} (binomial(2k,k)^2)*(4^(n-2*k))*T(n,k). - Bradley Klee, Feb 26 2018

A349154 Numbers k such that the k-th composition in standard order has sum equal to negative twice its alternating sum.

Original entry on oeis.org

0, 12, 160, 193, 195, 198, 204, 216, 240, 2304, 2561, 2563, 2566, 2572, 2584, 2608, 2656, 2752, 2944, 3074, 3077, 3079, 3082, 3085, 3087, 3092, 3097, 3099, 3102, 3112, 3121, 3123, 3126, 3132, 3152, 3169, 3171, 3174, 3180, 3192, 3232, 3265, 3267, 3270, 3276

Offset: 1

Gus Wiseman, Nov 21 2021

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The terms and corresponding compositions begin:
       0: ()
      12: (1,3)
     160: (2,6)
     193: (1,6,1)
     195: (1,5,1,1)
     198: (1,4,1,2)
     204: (1,3,1,3)
     216: (1,2,1,4)
     240: (1,1,1,5)
    2304: (3,9)
    2561: (2,9,1)
    2563: (2,8,1,1)
    2566: (2,7,1,2)
    2572: (2,6,1,3)
    2584: (2,5,1,4)

These compositions are counted by A224274 up to 0's.
Except for 0, a subset of A345919.
The positive version is A348614, reverse A349153.
An unordered version is A348617, counted by A001523.
The reverse version is A349155.
A positive unordered version is A349159, counted by A000712 up to 0's.
A000346 = even-length compositions with alt sum != 0, complement A001700.
A003242 counts Carlitz compositions.
A011782 counts compositions.
A025047 counts alternating or wiggly compositions, complement A345192.
A034871, A097805, and A345197 count compositions by alternating sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A116406 counts compositions with alternating sum >=0, ranked by A345913.
A138364 counts compositions with alternating sum 0, ranked by A344619.
Cf. A000070, A000984, A008549, A027306, A058622, A088218, A114121, A120452, A262977, A294175, A345917, A349160.
Statistics of standard compositions:
- The compositions themselves are the rows of A066099.
- Number of parts is given by A000120, distinct A334028.
- Sum and product of parts are given by A070939 and A124758.
- Maximum and minimum parts are given by A333766 and A333768.
Classes of standard compositions:
- Partitions and strict partitions are ranked by A114994 and A333256.
- Multisets and sets are ranked by A225620 and A333255.
- Strict and constant compositions are ranked by A233564 and A272919.
- Carlitz compositions are ranked by A333489, complement A348612.
- Necklaces are ranked by A065609, dual A333764, reversed A333943.
- Alternating compositions are ranked by A345167, complement A345168.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],Total[stc[#]]==-2*ats[stc[#]]&]

A007008 Chvatal conjecture for radius of graph of maximal intersecting sets.

Original entry on oeis.org

0, 1, 1, 3, 5, 11, 22, 47, 93, 193, 386, 793, 1586, 3238, 6476, 13167, 26333, 53381, 106762, 215955, 431910, 872218, 1744436, 3518265, 7036530, 14177066, 28354132, 57079714, 114159428, 229656076, 459312152, 923471727, 1846943453, 3711565741, 7423131482

Offset: 1

Daniel E. Loeb

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jan C. Bioch and Toshihide Ibaraki, Generating and approximating nondominated coteries, IEEE Transactions on parallel and distributed systems 6 (1995), 905-914.
D. E. Loeb and A. Meyerowitz, The maximal intersecting family of sets graph, in H. Barcelo and G. Kalai, editors, Proceedings of the Conference on Jerusalem Combinatorics 1993. AMS series Contemporary Mathematics, 1994. [broken link]
A. Meyerowitz, Maximal intersecting families, European J. Combin. 16 (1995), no. 5, 491-501.

Cf. A000346, A007007, A001206.

A007008(n)=2^n\4-binomial(n-1,(n-1)\2)\2 \\ - M. F. Hasler, Jan 14 2014

It is conjectured that a(2n+1)=A000346(n-1) for n>0. - Ralf Stephan, May 03 2004

a(n) = round(2^(n-2)-binomial(n-1,floor((n-1)/2))/2), cf. Thm. 14 in the Loeb-Meyerowitz paper. - M. F. Hasler, Jan 14 2014

A046527 A triangle related to A000108 (Catalan) and A000302 (powers of 4).

Original entry on oeis.org

1, 1, 1, 2, 5, 1, 5, 22, 9, 1, 14, 93, 58, 13, 1, 42, 386, 325, 110, 17, 1, 132, 1586, 1686, 765, 178, 21, 1, 429, 6476, 8330, 4746, 1477, 262, 25, 1, 1430, 26333, 39796, 27314, 10654, 2525, 362, 29, 1, 4862, 106762, 185517, 149052, 69930, 20754, 3973, 478, 33, 1

Offset: 0

Wolfdieter Lang

			Triangle begins as:
     1;
     1,      1;
     2,      5,      1;
     5,     22,      9,      1;
    14,     93,     58,     13,     1;
    42,    386,    325,    110,    17,     1;
   132,   1586,   1686,    765,   178,    21,    1;
   429,   6476,   8330,   4746,  1477,   262,   25,   1;
  1430,  26333,  39796,  27314, 10654,  2525,  362,  29,  1;
  4862, 106762, 185517, 149052, 69930, 20754, 3973, 478, 33,  1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Column sequences are: A000108 (k=0), A000346 (k=1), A018218 (k=2), A042941 (k=3), A042985 (k=4), A045505 (k=5), A045622 (k=6).

Row sums: A046814.

A046527:= func< n,k | k eq 0 select Catalan(n) else (1/2)*Binomial(n, k-1)*(4^(n-k+1) - Binomial(2*n, n)/(k*Catalan(k-1))) >;
[A046527(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 28 2024

T[n_, k_]:= If[k==0, CatalanNumber[n], (1/2)*Binomial[n,k-1]*(4^(n-k+ 1) -Binomial[2*n,n]/Binomial[2*(k-1),k-1])];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 28 2024 *)

def A046527(n,k):
    if k==0: return catalan_number(n)
    else: return (1/2)*binomial(n, k-1)*(4^(n-k+1) - binomial(2*n, n)/binomial(2*(k-1), k-1))
flatten([[A046527(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 28 2024

T(n, k) = binomial(n, k-1)*( 4^(n-k+1) - binomial(2*n, n)/binomial(2*(k-1), k-1) )/2, for n >= k >= 0, with T(n, 0) = A000108(n).

G.f. for column k: c(x)*(x/(1-4*x))^m, where c(x) = g.f. for Catalan numbers (A000108).

A094456 Triangle T(n,k), 0<=k<=n, read by rows; given by [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 5, 5, 0, 1, 10, 22, 14, 0, 1, 19, 70, 93, 42, 0, 1, 36, 201, 421, 386, 132, 0, 1, 69, 559, 1657, 2324, 1586, 429, 0, 1, 134, 1548, 6162, 11836, 12136, 6476, 1430, 0, 1, 263, 4316, 22445, 55843, 76928, 60948, 26333, 4862, 0, 1, 520, 12163, 81451, 254415, 444666, 467426, 297335, 106762, 16796

Offset: 0

Philippe Deléham, Jun 04 2004

Diagonals : A000007, A000012, A052944; A000108, A000346.
Triangle :
  1;
  0, 1;
  0, 1,  2;
  0, 1,  5,  5;
  0, 1, 10, 22, 14;
  ...
The alternating sum is (-1)^n = A033999(n). - F. Chapoton, Mar 18 2023

Alois P. Heinz, Rows n = 0..150, flattened

Cf. A000108 (main diagonal), A033999, A084938, A090365 (row sums).

Sum_{k=0..n} T(n,k) = A090365(n).

A258143 Row sums of A257241, Stifel's version of the arithmetical triangle.

Original entry on oeis.org

1, 2, 6, 10, 25, 41, 98, 162, 381, 637, 1485, 2509, 5811, 9907, 22818, 39202, 89845, 155381, 354521, 616665, 1401291, 2449867, 5546381, 9740685, 21977515, 38754731, 87167163, 154276027, 345994215, 614429671, 1374282018, 2448023842, 5461770405, 9756737701, 21717436833

Offset: 1

Wolfdieter Lang, May 22 2015

a(n) is the number of nonempty subsets of {1,2,...,n} that contain either more odd than even numbers or the same number of odd and even numbers. For example, for n=4, a(4)=10 and the 10 subsets are {1}, {3}, {1,3}, {1,2,3}, {1,3,4}; {1,2}, {1,4}, {2,3}, {3,4}, {1,2,3,4}. - Enrique Navarrete, Dec 16 2019

			n=3: a(3) = 2^3 - (1 + A008549(1)) = 8 - (1 + 1) = 6.
n=4: a(4) = 2^4 - (1 + A000346(1)) = 16 - (1 +  5) = 10.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A257241, A000346, A008549, A000108, A116406, A258144.

a258143 = sum . a257241_row  -- Reinhard Zumkeller, May 22 2015

Table[Sum[Binomial[n, m], {m, Ceiling[n/2]}], {n, 50}] (* Paolo Xausa, Nov 14 2024 *)

a(n) = Sum_{m = 1 .. ceiling(n/2)} binomial(n, m), n >= 1.
a(n) = 2^n - 2 - Sum_{i=1..floor(n/2)-1} binomial(n, i), n >= 2; a(1)=1. - Enrique Navarrete, Dec 16 2019
a(2*k+1) = 2^(2*k+1) - (1 + A008549(k)), k >= 0.
a(2*k) = 2^(2*k) - (1 + A000346(k-1)), k >= 1.
O.g.f.: x*(2+3*x+x^2 - (1-x^2)*(1+x)*c(x^2))/((1-(2*x)^2)*(1-x^2)) where c(x) is the o.g.f. of A000108.
O.g.f. for a(2*k+1), k >= 0: (2+x - (1-x)*c(x))/ ((1-4*x)*(1-x)).
O.g.f. for a(2*(k+1)), k >= 0: (3 - (1-x)*c(x))/ ((1-4*x)*(1-x)).
a(n) = A116406(n+1) - 1. - Hugo Pfoertner, Nov 14 2024

A346632 Triangle read by rows giving the main diagonals of the matrices counting integer compositions by length and alternating sum (A345197).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 0, 1, 2, 3, 0, 0, 0, 1, 2, 6, 6, 0, 0, 0, 1, 2, 9, 12, 0, 0, 0, 0, 1, 2, 12, 18, 10, 0, 0, 0, 0, 1, 2, 15, 24, 30, 20, 0, 0, 0, 0, 1, 2, 18, 30, 60, 60, 0, 0, 0, 0, 0, 1, 2, 21, 36, 100, 120, 35, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Jul 26 2021

The matrices (A345197) count the integer compositions of n of length k with alternating sum i, where 1 <= k <= n, and i ranges from -n + 2 to n in steps of 2. The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			Triangle begins:
   1
   0   0
   0   1   0
   0   1   2   0
   0   1   2   0   0
   0   1   2   3   0   0
   0   1   2   6   6   0   0
   0   1   2   9  12   0   0   0
   0   1   2  12  18  10   0   0   0
   0   1   2  15  24  30  20   0   0   0
   0   1   2  18  30  60  60   0   0   0   0
   0   1   2  21  36 100 120  35   0   0   0   0
   0   1   2  24  42 150 200 140  70   0   0   0   0
   0   1   2  27  48 210 300 350 280   0   0   0   0   0
   0   1   2  30  54 280 420 700 700 126   0   0   0   0   0

The first nonzero element in each column appears to be A001405.
These are the diagonals of the matrices given by A345197.
Antidiagonals of the same matrices are A345907.
Row sums are A345908.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices (reverse: A344616).
Other diagonals are A008277 of A318393 and A055884 of A320808.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000346, A007318, A008549, A025047, A163493, A344610.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],k==(n+ats[#])/2&]],{k,n}],{n,0,15}]

A349153 Numbers k such that the k-th composition in standard order has sum equal to twice its reverse-alternating sum.

Original entry on oeis.org

0, 11, 12, 14, 133, 138, 143, 148, 155, 158, 160, 168, 179, 182, 188, 195, 198, 204, 208, 216, 227, 230, 236, 240, 248, 2057, 2066, 2071, 2077, 2084, 2091, 2094, 2101, 2106, 2111, 2120, 2131, 2134, 2140, 2149, 2154, 2159, 2164, 2171, 2174, 2192, 2211, 2214

Offset: 1

Gus Wiseman, Nov 17 2021

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

			The terms and corresponding compositions begin:
    0: ()
   11: (2,1,1)
   12: (1,3)
   14: (1,1,2)
  133: (5,2,1)
  138: (4,2,2)
  143: (4,1,1,1,1)
  148: (3,2,3)
  155: (3,1,2,1,1)
  158: (3,1,1,1,2)
  160: (2,6)
  168: (2,2,4)
  179: (2,1,3,1,1)
  182: (2,1,2,1,2)
  188: (2,1,1,1,3)

These compositions are counted by A262977 up to 0's.
Except for 0, a subset of A345917.
The unreversed version is A348614.
The unreversed negative version is A349154.
The negative version is A349155.
A non-reverse unordered version is A349159, counted by A000712 up to 0's.
An unordered version is A349160, counted by A006330 up to 0's.
A003242 counts Carlitz compositions.
A011782 counts compositions.
A025047 counts alternating or wiggly compositions, complement A345192.
A034871, A097805, and A345197 count compositions by alternating sum.
A103919 counts partitions by alternating sum, reverse A344612.
A116406 counts compositions with alternating sum >=0, ranked by A345913.
A138364 counts compositions with alternating sum 0, ranked by A344619.
Cf. A000070, A000346, A001250, A001700, A008549, A027306, A058622, A088218, A114121, A120452, A294175.
Statistics of standard compositions:
- The compositions themselves are the rows of A066099.
- Number of parts is given by A000120, distinct A334028.
- Sum and product of parts are given by A070939 and A124758.
- Maximum and minimum parts are given by A333766 and A333768.
- Heinz number is given by A333219.
Classes of standard compositions:
- Partitions and strict partitions are ranked by A114994 and A333256.
- Multisets and sets are ranked by A225620 and A333255.
- Strict and constant compositions are ranked by A233564 and A272919.
- Carlitz compositions are ranked by A333489, complement A348612.
- Alternating compositions are ranked by A345167, complement A345168.

stc[n_]:=Differences[ Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[0,1000],Total[stc[#]]==2*sats[stc[#]]&]

A025135 (n-1)st elementary symmetric function of binomial(n,0), binomial(n,1), ..., binomial(n,n).

Original entry on oeis.org

1, 4, 22, 238, 5825, 345600, 51583084, 19765932032, 19661794008192, 51082239411000000, 347836712523276735000, 6221718604078720792473600, 292819054882445795002015111824, 36313083181879002042916296055971840, 11881691691176915544450299522846484375000

Offset: 1

Clark Kimberling

nonn

From R. J. Mathar, Oct 01 2016: (Start)
The k-th elementary symmetric functions of the terms binomial(n,j), j=0..n, form a triangle T(n,k), 0 <= k <= n, n >= 0:
  1
  1   2
  1   4    5
  1   8   22     24
  1  16   93    238     256
  1  32  386   2180    5825     6500
  1  64 1586  19184  117561   345600   407700
  1 128 6476 164864 2229206 15585920 51583084 64538880
  ...
This here is the first subdiagonal. The diagonal is A025134. The 2nd column is A000079, the 2nd A000346, the 3rd A025131, the 4th A025133. (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..70

a[n_] := SymmetricPolynomial[n-1, Table[Binomial[n, k], {k, 0, n}]]; a /@ Range[18] (* Jean-François Alcover, Jul 12 2011 *)

ESym(u)={my(v=vector(#u+1)); v[1]=1; for(i=1, #u, my(t=u[i]); forstep(j=i, 1,-1, v[j+1]+=v[j]*t)); v}
a(n)={ESym(binomial(n))[n]} \\ Andrew Howroyd, Dec 19 2018

cp's OEIS Frontend

A386567 a(n) = Sum_{k=0..n-1} binomial(6*k-1,k) * binomial(6*n-6*k,n-k-1).

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PARI

PARI

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A119245 Triangle, read by rows, defined by: T(n,k) = (4*k+1)*binomial(2*n+1, n-2*k)/(2*n+1) for n >= 2*k >= 0.

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Mathematica

PARI

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A349154 Numbers k such that the k-th composition in standard order has sum equal to negative twice its alternating sum.

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Mathematica

A007008 Chvatal conjecture for radius of graph of maximal intersecting sets.

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PARI

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A046527 A triangle related to A000108 (Catalan) and A000302 (powers of 4).

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Magma

Mathematica

SageMath

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A094456 Triangle T(n,k), 0<=k<=n, read by rows; given by [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is the operator defined in A084938.

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A258143 Row sums of A257241, Stifel's version of the arithmetical triangle.

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Haskell

Mathematica

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A346632 Triangle read by rows giving the main diagonals of the matrices counting integer compositions by length and alternating sum (A345197).

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A386567 a(n) = Sum_{k=0..n-1} binomial(6k-1,k) binomial(6n-6k,n-k-1).

A119245 Triangle, read by rows, defined by: T(n,k) = (4k+1)binomial(2n+1, n-2k)/(2n+1) for n >= 2k >= 0.