A024718 -id:A024718 - cp's OEIS frontend

A106268 Number triangle T(n,k) = (-1)^(n-k)binomial(k-n, n-k) = (0^(n-k) + binomial(2(n-k), n-k))/2 if k <= n, 0 otherwise; Riordan array (1/(2-C(x)), x) where C(x) is g.f. for Catalan numbers A000108.

1, 1, 1, 3, 1, 1, 10, 3, 1, 1, 35, 10, 3, 1, 1, 126, 35, 10, 3, 1, 1, 462, 126, 35, 10, 3, 1, 1, 1716, 462, 126, 35, 10, 3, 1, 1, 6435, 1716, 462, 126, 35, 10, 3, 1, 1, 24310, 6435, 1716, 462, 126, 35, 10, 3, 1, 1, 92378, 24310, 6435, 1716, 462, 126, 35, 10, 3, 1, 1

Offset: 0

Paul Barry, Apr 28 2005

Triangle includes A088218.

Inverse is A106270.

			Triangle (with rows n >= 0 and columns k >= 0) begins as follows:
    1;
    1,  1;
    3,  1,  1;
   10,  3,  1, 1;
   35, 10,  3, 1, 1;
  126, 35, 10, 3, 1, 1;
  ...
Production matrix begins:
    1, 1;
    2, 0, 1;
    5, 0, 0, 1;
   14, 0, 0, 0, 1;
   42, 0, 0, 0, 0, 1;
  132, 0, 0, 0, 0, 0, 1;
  429, 0, 0, 0, 0, 0, 0, 1;
  ... - _Philippe Deléham_, Oct 02 2014

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

Cf. A000108, A024718 (row sums), A088218, A106269 (diagonal sums), A106270.

A106268:= func< n,k | k eq n select 1 else (n-k+1)*Catalan(n-k)/2 >;
[A106268(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 10 2023

T[n_, k_]:= (-1)^(n-k)*Binomial[k-n, n-k];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 10 2023 *)

trg(nn) = {for (n=1, nn, for (k=1, n, print1(binomial(k-n,n-k)*(-1)^(n-k), ", ");); print(););} \\ Michel Marcus, Oct 03 2014

def A106268(n,k): return (1/2)*(0^(n-k) + (n-k+1)*catalan_number(n-k))
flatten([[A106268(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 10 2023

T(n, k) = (-1)^(n-k)*binomial(k-n, n-k).
T(n, k) = (1/2)*(0^(n-k) + binomial(2*(n-k), n-k)).
Sum_{k=0..n} T(n, k) = A024718(n) (row sums).
Sum_{k=0..floor(n/2)} T(n-k, k) = A106269(n) (diagonal sums).
Bivariate g.f.: Sum_{n, k >= 0} T(n,k)*x^n*y^k = (1/2) * (1/(1 - x*y)) * (1 + 1/sqrt(1 - 4*x)). - Petros Hadjicostas, Jul 15 2019

A121320 Number of vertices in all ordered (plane) trees with n edges that are at distance two from all the leaves above them.

Original entry on oeis.org

0, 0, 1, 2, 6, 18, 59, 203, 724, 2643, 9802, 36755, 138935, 528406, 2019419, 7748125, 29825844, 115132729, 445498768, 1727434607, 6710501025, 26110567532, 101744332967, 396983837719, 1550777652546, 6064476854065, 23739056348161

Offset: 0

Louis Shapiro, Aug 25 2006

			a(4)=6 since the root has the distance two property for the trees uudduudd and uudududd. There are similar points at height 1 for uuududdd, uuudddud and uduuuddd. The distance two point is at height 2 for uuuudddd.

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A000045, A024718.

CoefficientList[Series[x^2(1 + 1/Sqrt[1 - 4x])/(2(1 - x - x^2)), {x, 0, 26}], x] (* Robert G. Wilson v, Aug 21 2006 *)

seq(n)={Vec(x^2*(1 + 1/sqrt(1 - 4*x + O(x^(n-1))))/(2 - 2*x - 2*x^2), -(n+1))} \\ Andrew Howroyd, Apr 06 2020

G.f.: x^2*(1 + 1/sqrt(1 - 4*x))/(2 - 2*x - 2*x^2). - Reformulated by Georg Fischer, Apr 06 2020
Conjecture: (-n+2)*a(n) +(5*n-12)*a(n-1) +(-3*n+8)*a(n-2) +2*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Jun 22 2016
a(n) ~ 2^(2*n-1) / (11*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 07 2020

More terms from Robert G. Wilson v, Aug 21 2006

A122897 Riordan array (1/(1-x), c(x)-1) where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 8, 5, 1, 1, 22, 19, 7, 1, 1, 64, 67, 34, 9, 1, 1, 196, 232, 144, 53, 11, 1, 1, 625, 804, 573, 261, 76, 13, 1, 1, 2055, 2806, 2211, 1171, 426, 103, 15, 1, 1, 6917, 9878, 8399

Offset: 0

Paul Barry, Sep 18 2006

Product of A007318 and A122896. Inverse of Riordan array ((1+x+x^2)/(1+x)^2,x/(1+x)^2). Row sums are A024718.

The n-th row polynomial (in descending powers of x) equals the n-th Taylor polynomial of the rational function (1 - x^2)/(1 + x + x^2) * (1 + x)^(2*n) about 0. For example, for n = 4 we have (1 - x^2)/( 1 + x + x^2) * (1 + x)^8 = (x^4 + 22*x^3 + 19*x^2 + 7*x + 1) + O(x^5). - Peter Bala, Feb 21 2018

			Triangle begins
  1,
  1,     1,
  1,     3,     1,
  1,     8,     5,     1,
  1,    22,    19,     7,     1,
  1,    64,    67,    34,     9,    1,
  1,   196,   232,   144,    53,   11,    1,
  1,   625,   804,   573,   261,   76,   13,   1,
  1,  2055,  2806,  2211,  1171,  426,  103,  15,   1,
  1,  6917,  9878,  8399,  4979, 2126,  647, 134,  17,  1,
  1, 23713, 35072, 31655, 20483, 9878, 3554, 932, 169, 19, 1

P. Bala, A 4-parameter family of embedded Riordan arrays

A122897 := proc (n, k)
  binomial(2*n, n-k) + 2*add(cos((2/3)*Pi*j)*binomial(2*n, n-k-j), j = 1..n-k)
end proc:
for n from 0 to 10 do
seq(A122897(n, k), k = 0..n)
end do; # Peter Bala, Feb 21 2018

T(n,k) = binomial(2*n,n-k) + 2*Sum_{j = 1..n-k} cos((2/3)*Pi*j)* binomial(2*n, n-k-j). - Peter Bala, Feb 21 2018

T(n,k) = k*Sum_{i=0..n-k} C(2*(i+k),i)/(i+k), T(n,0)=1. - Vladimir Kruchinin, Jun 13 2020

A124642 Antidiagonal sums of A096465.

Original entry on oeis.org

1, 1, 2, 3, 5, 9, 15, 29, 50, 99, 176, 351, 638, 1275, 2354, 4707, 8789, 17577, 33099, 66197, 125477, 250953, 478193, 956385, 1830271, 3660541, 7030571, 14061141, 27088871, 54177741, 104647631, 209295261, 405187826, 810375651, 1571990936, 3143981871, 6109558586, 12219117171, 23782190486, 47564380971, 92705454896

Offset: 0

Gerald McGarvey, Dec 21 2006

nonn

Apparently bisections give A024718 and A006134 and are related to A078478, A100066 and A105848.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A006134, A024718, A078478, A100066, A105848.

a:= func< n | n eq 0 select 1 else (1+(-1)^n)/2 + (&+[ (&+[ ((n-2*j)/(n-2*k))*Binomial(n-2*k, n-k-j) : k in [0..j]]) : j in [0..Floor((n-1)/2)]]) >;
[a(n): n in [0..45]]; // G. C. Greubel, Apr 30 2021

a[, 0]=1; a[n, n_]=1; a[n_, m_]:= a[n, m] = a[n-1, m] + a[n, m-1]; a[n_, m_] /; n<0 || m>n = 0; Table[ Sum[a[n-m, m], {m,0,n}], {n,0,45}] (* Jean-François Alcover, Dec 17 2012 *)
a[n_]:= a[n]= (1+(-1)^n)/2 + Sum[(n-2*j)*Binomial[n-2*k, n-k-j]/(n-2*k), {j,0,(n-1)/2}, {k,0,j}]; Table[a[n], {n,0,45}] (* G. C. Greubel, Apr 30 2021 *)

def a(n): return (1+(-1)^n)/2 + sum( sum( ((n-2*j)/(n-2*k))*binomial(n-2*k, n-k-j) for k in (0..j)) for j in (0..(n-1)//2))
[a(n) for n in (0..45)] # G. C. Greubel, Apr 30 2021

Conjecture: G.f.: -(1/2)*z*(2*z+(1-4*z^2)^(1/2)+1)/(1-4*z^2)^(1/2)/(z^2-1). -  Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009
From G. C. Greubel, Apr 30 2021: (Start)
a(n) = (1 + (-1)^n)/2 + Sum_{j=0..floor((n-1)/2)} Sum_{k=0..j} (n-2*j)*binomial(n -2*k, n-k-j)/(n-2*k).
a(n) = Sum_{j=0..floor(n/2)} Sum_{k=0..j} ((n-2*j)/(n-k-j))*binomial(n-2*k, n-k-j). (End)

Offset changed by Reinhard Zumkeller, Jul 12 2012

Terms a(18) onward added by G. C. Greubel, Apr 30 2021

A360211 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(2n-3k,n-2*k).

Original entry on oeis.org

1, 2, 5, 17, 61, 221, 812, 3021, 11344, 42899, 163146, 623320, 2390653, 9198879, 35494701, 137290466, 532149805, 2066501909, 8038146035, 31312535610, 122140123201, 477002869614, 1864912495716, 7298427590543, 28588888586743, 112080607196843, 439744801379594

Offset: 0

Seiichi Manyama, Jan 30 2023

nonn

Cf. A005317, A024718, A176332, A360185.
Cf. A000108, A176287.
Cf. A026641, A360212.

A360211 := proc(n)
    add((-1)^k*binomial(2*n-3*k,n-2*k),k=0..floor(n/2)) ;
end proc:
seq(A360211(n),n=0..40) ; # R. J. Mathar, Mar 02 2023

a(n) = sum(k=0, n\2, (-1)^k*binomial(2*n-3*k, n-2*k));

my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1+2*x^2/(1+sqrt(1-4*x)))))

G.f.: 1 / ( sqrt(1-4*x) * (1 + x^2 * c(x)) ), where c(x) is the g.f. of A000108.
a(n) ~ 2^(2*n+3) / (9*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 18 2023
D-finite with recurrence 2*n*a(n) +(-5*n+2)*a(n-1) +(-11*n+12)*a(n-2) +2*(-n+5)*a(n-3) +(-7*n+2)*a(n-4) +2*(-2*n+5)*a(n-5)=0. - R. J. Mathar, Mar 02 2023

A371785 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(3n-k,n-2k).

Original entry on oeis.org

1, 3, 14, 76, 441, 2652, 16303, 101727, 641630, 4080154, 26112384, 167978615, 1085182436, 7035477777, 45750406205, 298279844724, 1949096816505, 12761551428024, 83701819019155, 549850618355886, 3617119500327536, 23824816811652905, 157106267803712709

Offset: 0

Seiichi Manyama, Apr 06 2024

nonn

Cf. A024718, A371786, A371787.

Cf. A371742.

a(n) = sum(k=0, n\2, (-1)^k*binomial(3*n-k, n-2*k));

a(n) = [x^n] 1/((1-x+x^2) * (1-x)^(2*n)).

A371786 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(4n-k,n-2k).

Original entry on oeis.org

1, 4, 27, 209, 1716, 14553, 125971, 1105885, 9809019, 87691592, 788832045, 7131655908, 64743390321, 589808771881, 5389066722654, 49365637128655, 453212161425716, 4168951499299185, 38415242186255419, 354527945536409116, 3276414018301664025

Offset: 0

Seiichi Manyama, Apr 06 2024

nonn

Cf. A024718, A371785, A371787.

Cf. A371743.

a(n) = sum(k=0, n\2, (-1)^k*binomial(4*n-k, n-2*k));

a(n) = [x^n] 1/((1-x+x^2) * (1-x)^(3*n)).

A371787 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(5n-k,n-2k).

Original entry on oeis.org

1, 5, 44, 441, 4675, 51129, 570401, 6451688, 73715212, 848793726, 9833394285, 114487194485, 1338411363535, 15700659542105, 184722993467063, 2178831068873601, 25756348168285379, 305061478075705411, 3619402085862708614, 43008294559624639777

Offset: 0

Seiichi Manyama, Apr 06 2024

Cf. A024718, A371785, A371786.

Cf. A371744.

a(n) = sum(k=0, n\2, (-1)^k*binomial(5*n-k, n-2*k));

a(n) = [x^n] 1/((1-x+x^2) * (1-x)^(4*n)).

It appears that a(n) = Sum_{k = 0..n} binomial(3*n+2*k-1, k). - Peter Bala, Jun 04 2024

A101475 Triangle T(n,k) read by rows: number of lattice paths from (0,0) to (0,2n) with steps (1,1) or (1,-1) that stay between the lines y=0 and y=k.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 15, 19, 20, 35, 50, 63, 69, 70, 126, 176, 217, 243, 251, 252, 462, 638, 770, 870, 913, 923, 924, 1716, 2354, 2794, 3159, 3355, 3419, 3431, 3432, 6435, 8789, 10307, 11610, 12430, 12766, 12855, 12869, 12870, 24310, 33099, 38489

Offset: 0

Ralf Stephan, Jan 21 2005

			Triangle begins
     1;
     1,    2;
     3,    5,     6;
    10,   15,    19,    20;
    35,   50,    63,    69,    70;
   126,  176,   217,   243,   251,   252;
   462,  638,   770,   870,   913,   923,   924;
  1716, 2354,  2794,  3159,  3355,  3419,  3431,  3432;
  6435, 8789, 10307, 11610, 12430, 12766, 12855, 12869, 12870;

W. Y. C. Cheng, E. Y. P. Deng, R. R. X. Du, R. P. Stanley and C. H. Yan, Crossings and nestings of matchings and partitions, arXiv:math/0501230 [math.CO], 2005.

Left-hand columns include A001700 and A024718. Right-hand columns include A000984 and A030662. Row sums are in A101476.

T[n_, k_] := Sum[Binomial[2n, n-i(k+2)] - Binomial[2n, n+i(k+2)+k+1], {i, 0, n}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 20 2019 *)

T(n, k) = Sum_{i>=0} (binomial(2n, n-i*(k+2)) - binomial(2n, n+i*(k+2)+k+1)).

A382225 Triangle read by rows: T(n,k) = Sum_{i=k..n} C(i-1,i-k)*C(i,k).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 7, 1, 1, 10, 25, 13, 1, 1, 15, 65, 73, 21, 1, 1, 21, 140, 273, 171, 31, 1, 1, 28, 266, 798, 871, 346, 43, 1, 1, 36, 462, 1974, 3321, 2306, 631, 57, 1, 1, 45, 750, 4326, 10377, 11126, 5335, 1065, 73, 1, 1, 55, 1155, 8646, 28017, 42878, 31795, 11145, 1693, 91, 1

Offset: 0

Vladimir Kruchinin, Mar 19 2025

Triangle T(n,k) of minors of the main diagonal of Pascal's matrix, n -size matrix, k - number of element of diagonal.

			Triangle starts:
  1;
  1,  1;
  1,  3,   1;
  1,  6,   7,   1;
  1, 10,  25,  13,   1;
  1, 15,  65,  73,  21,  1;
  1, 21, 140, 273, 171, 31, 1;
  ...

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).
Wikipedia, Minor (linear algebra)

Columns k=0-3 give: A000012, A000217, A001296(n-1) for n>=1, A107963(n-3) for n>=3.
Row sums give A024718.
T(n+1,n) gives A002061(n+1).
Cf. A007318, A184173.

T:= proc(n, k) option remember; `if`(n<0, 0,
      T(n-1, k)+binomial(n-1, k-1)*binomial(n, k))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Mar 20 2025

A382225[n_, k_] := A382225[n, k] = If[k == n, 1, A382225[n-1, k] + Binomial[n-1, k-1]*Binomial[n, k]];
Table[A382225[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Mar 22 2025 *)

h[i,j]:=binomial(i+j-3,i-1);
for n:1 thru 7 do
    if n=1 then print([1])
    else (M:genmatrix(h,n,n),
          print(makelist(determinant(minor(M,k,k)),k,1,n))
         );

G.f.: 1/(1-x) * ((1-x*(1-y))/(2*(sqrt((1-x*(1+y))^2-4*x^2*y)))+1/2).

T(n,k) = T(n-1,k)+C(n-1,k-1)*C(n,k).

cp's OEIS Frontend

A106268 Number triangle T(n,k) = (-1)^(n-k)*binomial(k-n, n-k) = (0^(n-k) + binomial(2*(n-k), n-k))/2 if k <= n, 0 otherwise; Riordan array (1/(2-C(x)), x) where C(x) is g.f. for Catalan numbers A000108.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A121320 Number of vertices in all ordered (plane) trees with n edges that are at distance two from all the leaves above them.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A122897 Riordan array (1/(1-x), c(x)-1) where c(x) is the g.f. of A000108.

Views

Author

Keywords

Comments

Examples

Links

Programs

Maple

Formula

A124642 Antidiagonal sums of A096465.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

A360211 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(2*n-3*k,n-2*k).

Views

Author

Keywords

Crossrefs

Programs

Maple

PARI

PARI

Formula

A371785 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(3*n-k,n-2*k).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A371786 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(4*n-k,n-2*k).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A371787 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(5*n-k,n-2*k).

Views

Author

Keywords

A106268 Number triangle T(n,k) = (-1)^(n-k)binomial(k-n, n-k) = (0^(n-k) + binomial(2(n-k), n-k))/2 if k <= n, 0 otherwise; Riordan array (1/(2-C(x)), x) where C(x) is g.f. for Catalan numbers A000108.

A360211 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(2n-3k,n-2*k).

A371785 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(3n-k,n-2k).

A371786 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(4n-k,n-2k).

A371787 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(5n-k,n-2k).