A119358 -id:A119358 - cp's OEIS frontend

A318557 Number A(n,k) of n-member subsets of [k*n] whose elements sum to a multiple of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 5, 10, 1, 0, 1, 1, 6, 30, 38, 1, 0, 1, 1, 9, 55, 165, 126, 1, 0, 1, 1, 10, 91, 460, 1001, 452, 1, 0, 1, 1, 13, 138, 969, 3876, 6198, 1716, 1, 0, 1, 1, 14, 190, 1782, 10630, 33594, 38760, 6470, 1, 0, 1, 1, 17, 253, 2925, 23751, 118755, 296010, 245157, 24310, 1, 0

Offset: 0

Alois P. Heinz, Aug 28 2018

The sequence of row n satisfies a linear recurrence with constant coefficients of order A018804(n) for n>0.

			A(3,2) = 10: {1,2,3}, {1,2,5}, {1,3,4}, {1,3,6}, {1,4,5}, {1,5,6}, {2,3,5}, {2,4,6}, {3,4,5}, {3,5,6}.
A(2,3) = 5: {1,2}, {1,5}, {2,4}, {3,6}, {4,5}.
Square array A(n,k) begins:
  1, 1,    1,     1,      1,       1,       1,        1, ...
  0, 1,    1,     1,      1,       1,       1,        1, ...
  0, 1,    2,     5,      6,       9,      10,       13, ...
  0, 1,   10,    30,     55,      91,     138,      190, ...
  0, 1,   38,   165,    460,     969,    1782,     2925, ...
  0, 1,  126,  1001,   3876,   10630,   23751,    46376, ...
  0, 1,  452,  6198,  33594,  118755,  324516,   749398, ...
  0, 1, 1716, 38760, 296010, 1344904, 4496388, 12271518, ...

Alois P. Heinz, Antidiagonals n = 0..85, flattened

Columns k=0-10 give: A000007, A000012, A119358, A318591, A318592, A318593, A318594, A318595, A318596, A318597, A318598.
Rows n=0-10 give: A000012, A057427, A042963 (for k>0), A318624, A318625, A318626, A318627, A318628, A318629, A318630, A318631.
Main diagonal gives A318477.
Cf. A018804, A304482.

nmax = 11; (* Program not suitable to compute a large number of terms. *)
A[n_, k_] := A[n, k] = Count[Subsets[Range[k n], {n}], s_ /; Divisible[Total[s], k]]; A[0, _] = 1;
Table[A[n - k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Oct 04 2019 *)

A282011 Number T(n,k) of k-element subsets of [n] having an even sum; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 4, 6, 3, 0, 1, 3, 6, 10, 9, 3, 0, 1, 3, 9, 19, 19, 9, 3, 1, 1, 4, 12, 28, 38, 28, 12, 4, 1, 1, 4, 16, 44, 66, 60, 40, 20, 5, 0, 1, 5, 20, 60, 110, 126, 100, 60, 25, 5, 0, 1, 5, 25, 85, 170, 226, 226, 170, 85, 25, 5, 1, 1, 6, 30, 110, 255, 396, 452, 396, 255, 110, 30, 6, 1

Offset: 0

Alois P. Heinz, Feb 04 2017

Row n is symmetric if and only if n mod 4 in {0,3} (or if T(n,n) = 1).

			T(5,0) = 1: {}.
T(5,1) = 2: {2}, {4}.
T(5,2) = 4: {1,3}, {1,5}, {2,4}, {3,5}.
T(5,3) = 6: {1,2,3}, {1,2,5}, {1,3,4}, {1,4,5}, {2,3,5}, {3,4,5}.
T(5,4) = 3: {1,2,3,4}, {1,2,4,5}, {2,3,4,5}.
T(5,5) = 0.
T(7,7) = 1: {1,2,3,4,5,6,7}.
Triangle T(n,k) begins:
  1;
  1, 0;
  1, 1,  0;
  1, 1,  1,   1;
  1, 2,  2,   2,   1;
  1, 2,  4,   6,   3,   0;
  1, 3,  6,  10,   9,   3,   0;
  1, 3,  9,  19,  19,   9,   3,   1;
  1, 4, 12,  28,  38,  28,  12,   4,   1;
  1, 4, 16,  44,  66,  60,  40,  20,   5,   0;
  1, 5, 20,  60, 110, 126, 100,  60,  25,   5,  0;
  1, 5, 25,  85, 170, 226, 226, 170,  85,  25,  5, 1;
  1, 6, 30, 110, 255, 396, 452, 396, 255, 110, 30, 6, 1;

Alois P. Heinz, Rows n = 0..200, flattened
Johann Cigler, Some remarks on Rogers-Szegö polynomials and Losanitsch's triangle, arXiv:1711.03340 [math.CO], 2017.
Johann Cigler, Some Pascal-like triangles, 2018.

Columns k=0..10 give (offsets may differ): A000012, A004526, A002620, A005993, A005994, A032092, A032093, A018211, A018212, A282077, A282078.
Row sums give A011782.
Main diagonal gives A133872(n+1).
Lower diagonals T(n+j,n) for j=1..10 give: A004525(n+1), A282079, A228705, A282080, A282081, A282082, A282083, A282084, A282085, A282086.
T(2n,n) gives A119358.
Cf. A007318, A057711, A087447, A159916.

b:= proc(n, s) option remember; expand(
      `if`(n=0, s, b(n-1, s)+x*b(n-1, irem(s+n, 2))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 1)):
seq(T(n), n=0..16);

Flatten[Table[Sum[Binomial[Ceiling[n/2],2j]Binomial[Floor[n/2],k-2j],{j,0,Floor[(n+1)/4]}],{n,0,10},{k,0,n}]] (* Indranil Ghosh, Feb 26 2017 *)

a(n,k)=sum(j=0,floor((n+1)/4),binomial(ceil(n/2),2*j)*binomial(floor(n/2),k-2*j));
tabl(nn)={for(n=0,nn,for(k=0,n,print1(a(n,k),", "););print(););} \\ Indranil Ghosh, Feb 26 2017

T(n,k) = Sum_{j=0..floor((n+1)/4)} C(ceiling(n/2),2*j) * C(floor(n/2),k-2*j).
T(n,k) = A007318(n,k) - A159916(n,k).
Sum_{k=0..n} k * T(n,k) = A057711(n-1) for n>0.
Sum_{k=0..n} (k+1) * T(n,k) = A087447(n) + [n=2].

A119326 Number triangle T(n,k) = Sum_{j=0..n-k} C(k,2j)*C(n-k,2j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 7, 10, 7, 1, 1, 1, 1, 11, 19, 19, 11, 1, 1, 1, 1, 16, 31, 38, 31, 16, 1, 1, 1, 1, 22, 46, 66, 66, 46, 22, 1, 1, 1, 1, 29, 64, 106, 126, 106, 64, 29, 1, 1, 1, 1, 37, 85, 162, 226, 226, 162, 85, 37, 1, 1

Offset: 0

Paul Barry, May 14 2006

Third column is essentially A000124. Fourth column is essentially A005448. Fifth column is A119327. Product of Pascal's triangle A007318 and A119328. Row sums are A038504. T(n,k) = T(n,n-k).

			Triangle begins:
  1;
  1, 1;
  1, 1,  1;
  1, 1,  1,  1;
  1, 1,  2,  1,  1;
  1, 1,  4,  4,  1,  1;
  1, 1,  7, 10,  7,  1, 1;
  1, 1, 11, 19, 19, 11, 1, 1;
  ...

Lukas Spiegelhofer and Jeffrey Shallit, Continuants, Run Lengths, and Barry's Modified Pascal Triangle, Volume 26(1) 2019, of The Electronic Journal of Combinatorics, #P1.31.

Seiichi Manyama, Rows n = 0..139, flattened
Jeffrey Shallit, Lukas Spiegelhofer, Continuants, run lengths, and Barry's modified Pascal triangle, arXiv:1710.06203 [math.CO], 2017.

Cf. A119358.

Column k has g.f.: (x^k/(1-x))* Sum{j=0..k} C(k,2j)*(x/(1-x))^(2j).

T(2n,n) = A119358(n). - Alois P. Heinz, Aug 31 2018

A169888 Number of n-member subsets of 1..2n whose elements sum to a multiple of n.

Original entry on oeis.org

1, 2, 2, 8, 18, 52, 152, 492, 1618, 5408, 18452, 64132, 225432, 800048, 2865228, 10341208, 37568338, 137270956, 504171584, 1860277044, 6892335668, 25631327688, 95640829924, 357975249028, 1343650267288, 5056424257552, 19073789328752, 72108867620204

Offset: 0

N. J. A. Sloane, Jul 07 2010, based on a letter from Jean-Claude Babois

nonn

This is twice A145855 (for n>0), which is the main entry for this problem.

Alois P. Heinz, Table of n, a(n) for n = 0..1669

Cf. A061865, A119358, A145855, A318431, A318432, A318433.

Column k=2 of A304482.

with(combinat): t0:=[]; for n from 1 to 8 do ans:=0; t1:=choose(2*n,n); for i in t1 do s1:=add(i[j],j=1..n); if s1 mod n = 0 then ans:=ans+1; fi; od: t0:=[op(t0),ans]; od:

a[n_] := Sum[(-1)^(n+d)*EulerPhi[n/d]*Binomial[2d, d]/n, {d, Divisors[n]}]; Table[a[n], {n, 1, 26}] (* Jean-François Alcover, Oct 22 2012, after T. D. Noe's program in A145855 *)

a(n) = if(n==0, 1, sumdiv(n, d, (-1)^(n+d)*eulerphi(n/d)*binomial(2*d, d)/n)); \\ Altug Alkan, Aug 27 2018, after T. D. Noe at A145855

a(n) = A061865(2n,n). - Alois P. Heinz, Aug 28 2018

a(n) ~ 2^(2*n) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 28 2023

a(0)=1 prepended by Alois P. Heinz, Aug 26 2018

A119363 a(n) = Sum_{k=0..n} C(n,3k)^2.

Original entry on oeis.org

1, 1, 1, 2, 17, 101, 402, 1275, 3921, 14114, 58601, 243695, 950578, 3537847, 13166791, 50514102, 198627921, 782913717, 3054480306, 11824753551, 45823049817, 178682390994, 700285942731, 2747647985241, 10767833451954, 42164261091351, 165225573240651

Offset: 0

Paul Barry, May 16 2006

a(n) - A119364(n) = A119365(n).

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Central coefficients of number triangle A119335.
a(n) = A119335(2n, n).
Cf. A024493, A119358, A139468, A139469.

Table[Sum[Binomial[n,3k]^2, {k,0,n}], {n,0,30}] (* Vaclav Kotesovec, Mar 12 2019 *)
Table[HypergeometricPFQ[{1/3 - n/3, 1/3 - n/3, 2/3 - n/3, 2/3 - n/3, -n/3, -n/3}, {1/3, 1/3, 2/3, 2/3, 1}, 1], {n, 0, 30}] (* Vaclav Kotesovec, Mar 12 2019 *)

From Vaclav Kotesovec, Mar 12 2019: (Start)
Recurrence: (n-2)*(n-1)*n*(637*n^6 - 11466*n^5 + 84364*n^4 - 324394*n^3 + 686227*n^2 - 755060*n + 336132)*a(n) = 3*(n-2)*(n-1)*(1274*n^7 - 23569*n^6 + 180194*n^5 - 733383*n^4 + 1699606*n^3 - 2208294*n^2 + 1449504*n - 351000)*a(n-1) - 3*(n-2)*(3185*n^8 - 63700*n^7 + 539028*n^6 - 2512118*n^5 + 7020469*n^4 - 11971242*n^3 + 12050010*n^2 - 6446736*n + 1362744)*a(n-2) + (14014*n^9 - 315315*n^8 + 3072678*n^7 - 16986046*n^6 + 58535088*n^5 - 129861691*n^4 + 184326992*n^3 - 159830656*n^2 + 75517728*n - 14313456)*a(n-3) + 3*(n-3)*(3185*n^8 - 63700*n^7 + 538391*n^6 - 2501394*n^5 + 6946794*n^4 - 11707256*n^3 + 11530544*n^2 - 5915328*n + 1142208)*a(n-4) + 18*(n-4)*(n-3)*(2*n - 9)*(637*n^6 - 7644*n^5 + 36589*n^4 - 88858*n^3 + 114124*n^2 - 71840*n + 16440)*a(n-5).
a(n) ~ 4^n / (3*sqrt(Pi*n)). (End)

Edited by N. J. A. Sloane, Jun 12 2008

A071688 Number of plane trees with even number of leaves.

Original entry on oeis.org

0, 1, 3, 7, 20, 66, 217, 715, 2424, 8398, 29414, 104006, 371384, 1337220, 4847637, 17678835, 64821680, 238819350, 883634026, 3282060210, 12233125112, 45741281820, 171529836218, 644952073662, 2430973096720, 9183676536076, 34766775829452, 131873975875180, 501121106988464

Offset: 1

Sen-peng Eu, Jun 23 2002

Number of Dyck n-paths with an even number of peaks (or, equivalently, odd number of valleys). - Yu Hin Au, Dec 07 2019

			a(3) = 3 because among the 5 plane 3-trees there are 3 trees with even number of leaves; a(4) = 7 because among the 14 plane 4-trees there are 7 trees with even number of leaves.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Yu Hin (Gary) Au, Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers, arXiv:1912.00555 [math.CO], 2019.
S. P. Eu, S. C. Liu and Y. N. Yeh, Odd or Even on Plane Trees, Discrete Mathematics, Volume 281, Issues 1-3, 28 April 2004, Pages 189-196.

a(n) + A071684 = A000108: Catalan numbers.

Cf. A007595.

[ &+[2*k*Binomial(n,2*k)^2/(n*(n-2*k+1)): k in [0..Floor(n/2)]] : n in [1..30]]; // G. C. Greubel, Dec 10 2019

seq( add(2*k*binomial(n,2*k)^2/(n*(n-2*k+1)), k=0..floor(n/2)), n=1..30); # G. C. Greubel, Dec 10 2019

a[n_] := If[EvenQ[n], Binomial[2n, n]/(2n + 2), Binomial[2n, n]/(2n + 2) + (-1)^((n + 1)/2)Binomial[n - 1, (n - 1)/2]/(n + 1)]
Table[(CatalanNumber[n] - 2^n Binomial[1/2, (n + 1)/2])/2, {n, 20}] (* Vladimir Reshetnikov, Oct 03 2016 *)

a(n) = 0^n + sum(k=1, n, (1/n)*binomial(n,k)*binomial(n,k-1)*(1+(-1)^k)/2); \\ Michel Marcus, Dec 09 2019

[ sum(2*k*binomial(n,2*k)^2/(n*(n-2*k+1)) for k in (0..floor(n/2))) for n in (1..30)] # G. C. Greubel, Dec 10 2019

a(2n) = (1/(4*n+2))*binomial(4*n, 2*n), a(2n+1) = (1/(4*n+4))*binomial(4*n+2, 2*n+1) + (-1)^(n+1)*(1/(2*n+2))*binomial(2*n, n).
G.f.: (1/4)*(2-(1-4*x)^(1/2) + 2*x - (1+4*x^2)^(1/2))/x. - Vladeta Jovovic, Apr 19 2003
a(0)=1, a(n) = Sum_{k=0..floor(n/2)} (1/n)*C(n,2k-1)*C(n,2k), n>0. - Paul Barry, Jan 25 2007
a(n) = 0^n + Sum_{k=1..n} (1/n)*C(n,k)*C(n,k-1)*(1+(-1)^k)/2. - Paul Barry, Dec 16 2008
a(n) = Sum_{k=0..n} Sum_{j=0..n-k} (-1)^j*(C(n,2*k)*C(n,2*k+j) - C(n,2*k-1)*C(n,2*k+j+1)). - Paul Barry, Sep 13 2010
n*(n+1)*a(n) -2*n*(n+1)*a(n-1) - 4*(2*n^2 -10*n +9)*a(n-2) +8*(n^2 -11*n + 21)*a(n-3) -48*(n-3)*(n-4)*a(n-4) + 32*(2*n-9)*(n-5)*a(n-5) = 0. - R. J. Mathar, Nov 24 2012 (corrected by Yu Hin Au, Dec 09 2019 )
a(n) = (A000108(n) - 2^n * binomial(1/2, (n+1)/2))/2. - Vladimir Reshetnikov, Oct 03 2016
From Vaclav Kotesovec, Oct 04 2016: (Start)
Recurrence (of order 3): n*(n+1)*(5*n^2 - 20*n + 18)*a(n) = 2*n*(2*n - 5)*(5*n^2 - 10*n + 3)*a(n-1) - 4*(n-2)*n*(5*n^2 - 20*n + 18)*a(n-2) + 8*(n-3)*(2*n - 5)*(5*n^2 - 10*n + 3)*a(n-3).
a(n) ~ 2^(2*n-1)/(sqrt(Pi*n)*n).
(End)
a(n) = A119358(n) - A119359(n) = hypergeom([1/2-n/2, 1/2-n/2, -n/2, -n/2], [1/2, 1/2, 1], 1) - hypergeom([-1/2-n/2, 1/2-n/2, 1-n/2, -n/2], [1/2, 1/2, 1], 1). - Vladimir Reshetnikov, Oct 05 2016

Edited by Robert G. Wilson v, Jun 25 2002

A318477 Number of n-member subsets of [n^2] whose elements sum to a multiple of n.

Original entry on oeis.org

1, 1, 2, 30, 460, 10630, 324516, 12271518, 553275192, 28987537806, 1731030733840, 116068178638786, 8634941165110140, 705873715441872276, 62895036883536770108, 6067037854078500844740, 629921975126483973659888, 70043473196734767582082246

Offset: 0

Alois P. Heinz, Aug 26 2018

nonn

			a(0) = 1: {}.
a(1) = 1: {1}.
a(2) = 2: {1,3}, {2,4}.
a(3) = 30: {1,2,3}, {1,2,6}, {1,2,9}, {1,3,5}, {1,3,8}, {1,4,7}, {1,5,6}, {1,5,9}, {1,6,8}, {1,8,9}, {2,3,4}, {2,3,7}, {2,4,6}, {2,4,9}, {2,5,8}, {2,6,7}, {2,7,9}, {3,4,5}, {3,4,8}, {3,5,7}, {3,6,9}, {3,7,8}, {4,5,6}, {4,5,9}, {4,6,8}, {4,8,9}, {5,6,7}, {5,7,9}, {6,7,8}, {7,8,9}.

Alois P. Heinz, Table of n, a(n) for n = 0..338

Main diagonal of A304482 and of A318557.

Cf. A119358, A169888, A308667.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(phi(n/d)*
      (-1)^(n+d)*binomial(n*d, d), d=divisors(n))/n)
    end:
seq(a(n), n=0..20);

a[n_] := (-1)^n Sum[(-1)^d Binomial[d n, d] EulerPhi[n/d], {d, Divisors[n]} ]/n; a[0] = 1;
a /@ Range[0, 20] (* Jean-François Alcover, Sep 23 2019 *)

a(n) = n * A308667(n) for n >= 1.

a(n) ~ exp(n - 1/2) * n^(n - 3/2) / sqrt(2*Pi). - Vaclav Kotesovec, Mar 28 2023

A307091 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n,2*k)^2.

Original entry on oeis.org

1, 1, 0, -8, -34, -74, 0, 736, 3334, 7606, 0, -80464, -372436, -864772, 0, 9400192, 43976774, 103061158, 0, -1137528688, -5355697084, -12623082284, 0, 140697113792, 665238165916, 1574005263676, 0, -17663830073504, -83769667651816, -198760191043784, 0

Offset: 0

Seiichi Manyama, Mar 24 2019

Seiichi Manyama, Table of n, a(n) for n = 0..1879

Central coefficients of number triangle A307090.

Cf. A000984, A119358, A119363.

Table[Sum[(-1)^k*Binomial[n, 2*k]^2, {k, 0, Floor[n/2]}], {n, 0, 30}] (* Vaclav Kotesovec, Mar 24 2019 *)
Table[HypergeometricPFQ[{1/2 - n/2, 1/2 - n/2, -n/2, -n/2}, {1/2, 1/2, 1}, -1], {n, 0, 30}] (* Vaclav Kotesovec, Mar 24 2019 *)

{a(n) = sum(k=0, n\2, (-1)^k*binomial(n, 2*k)^2)}

a(4*n+2) = 0 for n >= 0.
From Peter Bala, Mar 17 2023: (Start)
n*(n-1)*(6*n^2-24*n+23)a(n) = 4*(n-1)*(2*n-3)*(3*n^2-9*n+4)*a(n-1) - 4*(3*n^2-9*n+4)*(2*n-3)^2*a(n-2) - 8*(n-2)*(2*n-3)*(3*n^2-9*n+4)*a(n-3) - 4*(n-2)*(n-3)*(6*n^2-12*n+5)*a(n-4) with a(0) = 1, a(1) = 1, a(2) = 0 and a(3) = -8.
a(n) = hypergeom([(1-n)/2, (1-n)/2, -n/2, -n/2], [1/2, 1/2, 1], -1).
Conjecture: the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(2*r)) holds for positive integers n and r and all primes p >= 3. (End)

A119359 Central coefficients of number triangle A119326.

Original entry on oeis.org

0, 1, 1, 7, 31, 106, 386, 1499, 5755, 21886, 83854, 323302, 1248534, 4828916, 18719364, 72711123, 282867795, 1101981430, 4298723990, 16788997874, 65641296578, 256895812108, 1006307847324, 3945185527582, 15478851119966

Offset: 0

Paul Barry, May 16 2006

a(n) = A119326(2n,n+1). A119358(n)-a(n) = A071688(n).

Table[HypergeometricPFQ[{-1/2 - n/2, 1/2 - n/2, 1 - n/2, -n/2}, {1/2, 1/2, 1}, 1] - KroneckerDelta[n], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 04 2016 *)
Table[(2^n Binomial[1/2, (n+1)/2]  + Binomial[n, n/2] Cos[Pi n/2] + n CatalanNumber[n])/2 - KroneckerDelta[n], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 06 2016 *)

G.f.: (1/sqrt(1-4x)+(1/sqrt(1+4x^2)-1)-c(x)+x*c(-x^2))/2, c(x) the g.f. of A000108;
a(n) = (C(2n,n+1)+C((n-1)/2)*sin(Pi*n/2)-2*0^n-2C(n-1,n/2)*sin(Pi*(n-1)/2))/2.
a(n) = hypergeom([-1/2-n/2, 1/2-n/2, 1-n/2, -n/2], [1/2, 1/2, 1], 1) - 0^n. - Vladimir Reshetnikov, Oct 04 2016

cp's OEIS Frontend

A318557 Number A(n,k) of n-member subsets of [k*n] whose elements sum to a multiple of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A282011 Number T(n,k) of k-element subsets of [n] having an even sum; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A119326 Number triangle T(n,k) = Sum_{j=0..n-k} C(k,2j)*C(n-k,2j).

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Formula

A169888 Number of n-member subsets of 1..2n whose elements sum to a multiple of n.

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Maple

Mathematica

PARI

Formula

Extensions

A119363 a(n) = Sum_{k=0..n} C(n,3k)^2.

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Mathematica

Formula

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A071688 Number of plane trees with even number of leaves.

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Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A318477 Number of n-member subsets of [n^2] whose elements sum to a multiple of n.

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