A100324 -id:A100324 - cp's OEIS frontend

A100326 Triangle, read by rows, where row n equals the inverse binomial of column n of square array A100324, which lists the self-convolutions of SHIFT(A003169).

1, 1, 1, 3, 4, 1, 14, 20, 7, 1, 79, 116, 46, 10, 1, 494, 736, 311, 81, 13, 1, 3294, 4952, 2174, 626, 125, 16, 1, 22952, 34716, 15634, 4798, 1088, 178, 19, 1, 165127, 250868, 115048, 36896, 9094, 1724, 240, 22, 1, 1217270, 1855520, 862607, 285689, 74687, 15629, 2561, 311, 25, 1

Offset: 0

Paul D. Hanna, Nov 17 2004

The leftmost column equals A003169 shift one place right.
Each column k > 0 equals the convolution of the prior column and A003169.
Row sums form A100327.
The elements of the matrix inverse are T^(-1)(n,k) = (-1)^(n+k) * A158687(n,k). - R. J. Mathar, Mar 15 2013

			Rows begin:
        1;
        1,       1;
        3,       4,      1;
       14,      20,      7,      1;
       79,     116,     46,     10,     1;
      494,     736,    311,     81,    13,     1;
     3294,    4952,   2174,    626,   125,    16,    1;
    22952,   34716,  15634,   4798,  1088,   178,   19,   1;
   165127,  250868, 115048,  36896,  9094,  1724,  240,  22,  1;
  1217270, 1855520, 862607, 285689, 74687, 15629, 2561, 311, 25,  1;
  ...
First column forms A003169 shift right.
Binomial transform of row 3 forms column 3 of square A100324: BINOMIAL([14,20,7,1]) = [14,34,61,96,140,194,259,...].
Binomial transform of row 4 forms column 4 of square A100324: BINOMIAL([79,116,46,10,1]) = [79,195,357,575,860,1224,...].

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Cf. A003169, A100324, A100327 (row sums), A158687, A264717 (central terms).

import Data.List (transpose)
a100326 n k = a100326_tabl !! n !! k
a100326_row n = a100326_tabl !! n
a100326_tabl = [1] : f [[1]] where
f xss@(xs:_) = ys : f (ys : xss) where
ys = y : map (sum . zipWith (*) (zs ++ [y])) (map reverse zss)
y = sum $ zipWith (*) [1..] xs
zss@((:zs):) = transpose $ reverse xss
-- Reinhard Zumkeller, Nov 21 2015

A100326 := proc(n,k)
    if k < 0 or k > n then
        0 ;
    elif n = 0 then
        1 ;
    elif k = 0 then
        A003169(n)
    else
        add(procname(i+1,0)*procname(n-i-1,k-1),i=0..n-k) ;
    end if;
end proc: # R. J. Mathar, Mar 15 2013

lim= 9; t[0, 0]=1; t[n_, 0]:= t[n, 0]= Sum[(k+1)*t[n-1,k], {k,0,n-1}]; t[n_, k_]:= t[n, k]= Sum[t[j+1, 0]*t[n-j-1, k-1], {j,0,n-k}]; Flatten[Table[t[n, k], {n,0,lim}, {k,0,n}]] (* Jean-François Alcover, Sep 20 2011 *)

PARI
```
T(n,k)=if(n
				
```

@CachedFunction
def T(n,k): # T = A100326
    if (k<0 or k>n): return 0
    elif (k==n): return 1
    elif (k==0): return sum((j+1)*T(n-1,j) for j in range(n))
    else: return sum(T(j+1,0)*T(n-j-1,k-1) for j in range(n-k+1))
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 30 2023

T(n, 0) = A003169(n) = Sum_{k=0..n-1} (k+1)*T(n-1, k) for n>0, with T(0, 0)=1.
T(n, k) = Sum_{i=0..n-k} T(i+1, 0)*T(n-i-1, k-1) for n > 0.
T(2*n, n) = A264717(n).
Sum_{k=0..n} T(n, k) = A100327(n).
G.f.: A(x, y) = (1 + G(x))/(1 - y*G(x)), where G(x) is the g.f. of A003169.
From G. C. Greubel, Jan 30 2023: (Start)
Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n).
Sum_{k=0..n-1} (-1)^k*T(n, k) = A033999(n). (End)

A100327 Row sums of triangle A100326, in which row n equals the inverse binomial of column n of square array A100324.

Original entry on oeis.org

1, 2, 8, 42, 252, 1636, 11188, 79386, 579020, 4314300, 32697920, 251284292, 1953579240, 15336931928, 121416356108, 968187827834, 7769449728780, 62696580696172, 508451657412496, 4141712433518956, 33872033298518728, 278014853384816184, 2289376313410678312

Offset: 0

Paul D. Hanna, Nov 17 2004

nonn

Self-convolution yields A100328, which equals column 1 of triangle A100326 (omitting leading zero).

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A003168, A003169, A100326, A219538.

A100327:= func< n | n eq 0 select 1 else (2/n)*(&+[Binomial(n, k)*Binomial(2*n+k, k-1): k in [1..n]]) >;
[A100327(n): n in [0..30]]; // G. C. Greubel, Jan 30 2023

A100327 := n -> simplify(2^n*binomial(3*n,2*n)*hypergeom([-1-2*n,-n], [-3*n], 1/2)/ (n+1/2)): seq(A100327(n), n=0..22); # Peter Luschny, Jun 10 2017

Flatten[{1,Table[Sum[2*Binomial[n,k]*Binomial[2n+k,k-1]/n,{k,1,n}],{n,1,20}]}] (* Vaclav Kotesovec, Oct 17 2012 *)

a(n)=if(n==0,1,sum(k=0,n,2*binomial(n,k)*binomial(2*n+k,k-1)/n))

a(n)=polcoeff((1/x)*serreverse(x*(1-x+sqrt(1-4*x +x^2*O(x^n)))/(2+x)),n)
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Nov 22 2012

def A100327(n): return 2^n*binomial(3*n,2*n)*simplify(hypergeometric([-1-2*n,-n], [-3*n],1/2)/(n+1/2))
[A100327(n) for n in range(31)] # G. C. Greubel, Jan 30 2023

G.f.: (1/x)*Series_Reversion( x*(1-x + sqrt(1 - 4*x)) / (2+x) ). - Paul D. Hanna, Nov 22 2012
G.f. A(x) = (1+G(x))/(1-G(x)), also A(x)^2 = (1+G(x))*G(x)/x, where G(x) = x*(1+G(x))/(1-G(x))^2 is the g.f. of A003169.
a(n) = 2*A003168(n) for n>0 with a(0)=1.
a(n) = Sum_{k=1..n} 2*binomial(n, k)*binomial(2n+k, k-1)/n for n>0 with a(0)=1.
Recurrence: 20*n*(2*n+1)*a(n) = (371*n^2 - 395*n + 96)*a(n-1) - 6*(27*n^2 - 103*n + 96)*a(n-2) + 4*(n-3)*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ sqrt(4046 + 1122*sqrt(17))*((71 + 17*sqrt(17))/16)^n/(136*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012
a(n) = 2^n*binomial(3*n,2*n)*hypergeometric([-1-2*n,-n], [-3*n],1/2)/(n+1/2). - Peter Luschny, Jun 10 2017

A100325 Antidiagonal sums of square array A100324, which lists the self-convolutions of SHIFT(A003169).

Original entry on oeis.org

1, 2, 6, 25, 130, 774, 5009, 34231, 242988, 1773767, 13229272, 100362848, 772016385, 6007208105, 47198747457, 373929821070, 2983774582206, 23958802697161, 193448157014605, 1569625544848531, 12791865082236462

Offset: 0

Paul D. Hanna, Nov 17 2004

nonn

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

Cf. A003169, A100324.

f[n_]:= f[n]= If[n<2, 1, If[n==2, 3, ((324*n^2 -708*n +360)*f[n-1] -(371*n^2 -1831*n +2250)*f[n-2] + (20*n^2 -130*n +210)*f[n-3])/(16*n*(2*n-1)) ]]; (* f = A003169 *)
A[n_, k_]:= A[n, k]= If[n==0, f[k], If[k==0, 1, Sum[f[k-j]*A[n-1,j], {j,0,k}]]]; (* A = 100324 *)
a[n_]:= a[n]= Sum[A[n-k,k], {k,0,n}]; (* a = A100325 *)
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Jan 31 2023 *)

{a(n)=local(A=1+x+x*O(x^n));if(n==0,1, for(i=1,n,A=1+x*A/(2-A)^2); sum(k=0,n,polcoeff(A^(n-k+1),k)))}

@CachedFunction
def f(n): # f = A003169
    if (n<2): return 1
    elif (n==2): return 3
    else: return ((324*n^2-708*n+360)*f(n-1) - (371*n^2-1831*n+2250)*f(n-2) + (20*n^2-130*n+210)*f(n-3))/(16*n*(2*n-1))
@CachedFunction
def A(n, k): # A = 100324
    if (n==0): return f(k)
    elif (k==0): return 1
    else: return sum( f(k-j)*A(n-1, j) for j in range(k+1) )
def T(n,k): return A(n-k, k)
def A100325(n): return sum( T(n,k) for k in range(n+1) )
[A100325(n) for n in range(41)] # G. C. Greubel, Jan 31 2023

G.f. A(x) = (1+G003169(x))/(1-x-x*G003169(x)), where G003169(x) is the g.f. of A003169.

a(n) ~ (sqrt(3056686 + 12607266/sqrt(17)) * ((71 + 17*sqrt(17))/16)^n) / (10201 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jan 31 2023

A100328 Column 1 of triangle A100326, in which row n equals the inverse binomial of column n of square array A100324, with leading zero omitted.

Original entry on oeis.org

1, 4, 20, 116, 736, 4952, 34716, 250868, 1855520, 13979192, 106901032, 827644424, 6474611984, 51100656544, 406400018092, 3253636464756, 26201323746880, 212093247874904, 1724793778005528, 14084738953391768, 115447965121881856

Offset: 0

Paul D. Hanna, Nov 17 2004

Self-convolution of A100327, which equals the row sums of triangle A100326.

Vaclav Kotesovec, Table of n, a(n) for n = 0..650

Cf. A003168, A003169, A100324, A100326, A100327.

{a(n)=if(n==0,1,sum(j=0,n, if(j==0,1,sum(k=0,j,2*binomial(j,k)*binomial(2*j+k,k-1)/j))* if(n-j==0,1,sum(k=0,n-j,2*binomial(n-j,k)*binomial(2*n-2*j+k,k-1)/(n-j)))))}

G.f.: (1+G003169(x))*G003169(x)/x, where G003169(x) is the g.f. of A003169.
Recurrence: 4*(n+1)*(2*n+1)*(17*n^2 - 28*n + 12)*a(n) = (1207*n^4 - 1988*n^3 + 1013*n^2 - 124*n - 12)*a(n-1) - 2*(n-2)*(2*n-3)*(17*n^2 + 6*n + 1)*a(n-2). - Vaclav Kotesovec, Jul 05 2014
a(n) ~ sqrt(95+393/sqrt(17)) * ((71+17*sqrt(17))/16)^n / (4*sqrt(2*Pi) * n^(3/2)). - Vaclav Kotesovec, Jul 05 2014
From Peter Bala, Sep 08 2024: (Start)
a(n) = (2/n) * Sum_{k = 0..n} binomial(n+1, n-k-1)*binomial(2*n, k)*2^(n-k) for n >= 1.
a(n) = 4*Jacobi_P(n-1, 2, n+1, 3)/n for n >= 1. Cf. A003168. (End)

A003169 Number of 2-line arrays; or number of P-graphs with 2n edges.

Original entry on oeis.org

1, 3, 14, 79, 494, 3294, 22952, 165127, 1217270, 9146746, 69799476, 539464358, 4214095612, 33218794236, 263908187100, 2110912146295, 16985386737830, 137394914285538, 1116622717709012, 9113225693455362, 74659999210200292

Offset: 1

N. J. A. Sloane

First column of triangle A100326. - Paul D. Hanna, Nov 16 2004

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

Vincenzo Librandi, Table of n, a(n) for n = 1..200
M. Bicknell and V. E. Hoggatt, Jr., Sequences of matrix inverses from Pascal, Catalan and related convolution arrays, Fib. Quart., 14 (1976), 224-232.
L. Carlitz, Enumeration of two-line arrays, Fib. Quart., Vol. 11 Number 2 (1973), 113-130.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 416
R. C. Read, On the enumeration of a class of plane multigraphs, Aequat. Math. 31 (1986) no 1, 47-63.
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 8.
Anssi Yli-Jyrä and Carlos Gómez-Rodríguez, Generic Axiomatization of Families of Noncrossing Graphs in Dependency Parsing, arXiv:1706.03357 [cs.CL], 2017.

Cf. A003168, A100324, A100326, A156894.

a003169 = flip a100326 0  -- Reinhard Zumkeller, Nov 21 2015

a[0]:=0:a[1]:=1:a[2]:=3:for n from 3 to 30 do a[n]:=((324*n^2-708*n+360)*a[n-1] -(371*n^2-1831*n+2250)*a[n-2]+(20*n^2-130*n+210)*a[n-3])/(16*n*(2*n-1)) od:seq(a[n],n=1..25); # Emeric Deutsch, Jan 31 2005

lim = 21; t[0, 0] = 1; t[n_, 0] := t[n, 0] = Sum[(k + 1)*t[n - 1, k], {k, 0, n - 1}]; t[n_, k_] := t[n, k] = Sum[t[j + 1, 0]*t[n - j - 1, k - 1], {j, 0, n - k}]; Table[ t[n, 0], {n, lim}] (* Jean-François Alcover, Sep 20 2011, after Paul D. Hanna's comment *)

{a(n)=if(n==0,0,if(n==1,1,if(n==2,3,( (324*n^2-708*n+360)*a(n-1) -(371*n^2-1831*n+2250)*a(n-2)+(20*n^2-130*n+210)*a(n-3))/(16*n*(2*n-1)) )))} \\ Paul D. Hanna, Nov 16 2004

{a(n)=local(A=x+x*O(x^n));if(n==1,1, for(i=1,n,A=x*(1+A)/(1-A)^2); polcoeff(A,n))}

seq(n)=Vec(serreverse(x*(1 - x)^2/(1 + x) + O(x*x^n))) \\ Andrew Howroyd, Mar 07 2023

For formula see Read reference.
D-finite with recurrence a(n) = ( (324*n^2-708*n+360)*a(n-1) - (371*n^2-1831*n+2250)*a(n-2) + (20*n^2-130*n+210)*a(n-3) )/(16*n*(2*n-1)) for n>2, with a(0)=0, a(1)=1, a(2)=3. - Paul D. Hanna, Nov 16 2004
G.f. satisfies: A(x) = x*(1+A(x))/(1-A(x))^2 where A(0)=0. G.f. satisfies: (1+A(x))/(1-A(x)) = 2*G003168(x)-1, where G003168 is the g.f. of A003168. - Paul D. Hanna, Nov 16 2004
a(n) = (1/n)*Sum_{i=0..n-1} binomial(n,i)*binomial(3*n-i-2,n-i-1). - Vladeta Jovovic, Sep 13 2006
Appears to be (1/n)*Jacobi_P(n-1,1,n-1,3). If so then a(n) = (1/(2*n-1))*Sum_{k = 0..n-1} binomial(n-1,k)*binomial(2*n+k-1,k+1) = (1/n)*Sum_{k = 0..n} binomial(n,k)*binomial(2*n-2,n+k-1)*2^k. [Added Jun 11 2023: these are correct, and can be proved using the WZ algorithm.] - Peter Bala, Aug 01 2012
a(n) ~ sqrt(33/sqrt(17)-7) * ((71+17*sqrt(17))/16)^n / (4*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 09 2013
The o.g.f. A(x) = 1 + 3*x + 14*x^2 + ... taken with offset 0, satisfies 1 + x*A'(x)/A(x) = 1 + 3*x + 19*x^2 + 138*x^3 + ..., the o.g.f. for A156894. - Peter Bala, Oct 05 2015
From Peter Bala, Jun 11 2023: (Start)
a(n) = (1/n)*Sum_{k = 0..n-1} binomial(n,k+1)*binomial(2*n+k-1,k) (Carlitz, equation 3.19).
4*n*(17*n - 29)*(2*n - 1)*a(n) = (1207*n^3 - 4473*n^2 + 5258*n - 1920)*a(n-1) - 2*(2*n - 5)*(17*n - 12)*(n - 2)*a(n-2) with a(1) = 1 and a(2) = 3. (End)

More terms from Emeric Deutsch, Jan 31 2005

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A100326 Triangle, read by rows, where row n equals the inverse binomial of column n of square array A100324, which lists the self-convolutions of SHIFT(A003169).

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A100327 Row sums of triangle A100326, in which row n equals the inverse binomial of column n of square array A100324.

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A100325 Antidiagonal sums of square array A100324, which lists the self-convolutions of SHIFT(A003169).

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A100328 Column 1 of triangle A100326, in which row n equals the inverse binomial of column n of square array A100324, with leading zero omitted.

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A003169 Number of 2-line arrays; or number of P-graphs with 2n edges.

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