A128322 -id:A128322 - cp's OEIS frontend

A128320 Triangle, read by rows, where T(n,k) equals the dot product of the vector of terms in row n that are to the right of T(n,k) with the vector of terms in column k that are above T(n,k) for n>k+1>0, with the odd numbers in the secondary diagonal and all 1's in the main diagonal.

1, 1, 1, 4, 3, 1, 17, 8, 5, 1, 98, 41, 12, 7, 1, 622, 234, 73, 16, 9, 1, 4512, 1602, 418, 113, 20, 11, 1, 35373, 11976, 3110, 650, 161, 24, 13, 1, 300974, 98541, 23920, 5242, 930, 217, 28, 15, 1, 2722070, 866942, 207549, 41304, 8094, 1258, 281, 32, 17, 1

Offset: 0

Paul D. Hanna, Feb 25 2007

			Illustrate the recurrence by:
  T(n,k) = [T(n,k+1),T(n,k+2), ..,T(n,n)]*[T(k,k),T(k+1,k),..,T(n-1,k)]:
  T(3,0) = [8,5,1]*[1,1,4]~ = 8*1 + 5*1 + 1*4 = 17;
  T(4,1) = [12,7,1]*[1,3,8]~ = 12*1 + 7*3 + 1*8 = 41;
  T(5,1) = [73,16,9,1]*[1,3,8,41]~ = 73*1 + 16*3 + 9*8 + 1*41 = 234;
  T(6,2) = [113,20,11,1]*[1,5,12,73]~ = 113*1 + 20*5 + 11*12 + 1*73 = 418.
Triangle begins:
         1;
         1,       1;
         4,       3,       1;
        17,       8,       5,      1;
        98,      41,      12,      7,     1;
       622,     234,      73,     16,     9,     1;
      4512,    1602,     418,    113,    20,    11,    1;
     35373,   11976,    3110,    650,   161,    24,   13,   1;
    300974,   98541,   23920,   5242,   930,   217,   28,  15,  1;
   2722070,  866942,  207549,  41304,  8094,  1258,  281,  32, 17,  1;
  26118056, 8139602, 1885166, 377757, 65088, 11762, 1634, 353, 36, 19, 1;

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

Columns k: A128321 (k=0), A128322 (k=1), A128323 (k=2).
Sums: A128324 (row sums).
Variant of: A115080.

function T(n,k) // T = A128320
   if k eq n then return 1;
   elif k eq n-1 then return 2*n-1;
   else return (&+[T(n, k+j+1)*T(k+j, k): j in [0..n-k-1]]);
   end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 25 2024

T[n_, k_]:= T[n, k]= If[k==n, 1, If[k==n-1, 2*n-1, Sum[T[n,k+j+1] *T[k+j,k], {j,0,n-k-1}]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 25 2024 *)

{T(n,k)=if(n==k,1, if(n==k+1,2*n-1, sum(i=0,n-k-1, T(n,k+i+1)*T(k+i,k))))};
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))

@CachedFunction
def T(n,k): # T = A128320
    if k==n: return 1
    elif k==n-1: return 2*n-1
    else: return sum(T(n, k+j+1)*T(k+j, k) for j in range(n-k))
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jun 25 2024

T(n,k) = Sum_{j=0..n-1-k} T(n,k+j+1)*T(k+j,k) for n > k+1 > 0, with T(n,n) = 1 and T(n, n-1) = 2*n-1 for k >= 0.

A128321 Column 0 of triangle A128320.

Original entry on oeis.org

1, 1, 4, 17, 98, 622, 4512, 35373, 300974, 2722070, 26118056, 263266346, 2780054884, 30586452652, 349724463584, 4141218303165, 50678688359190, 639387728054310, 8302396672724280, 110754894628585950

Offset: 0

Paul D. Hanna, Feb 25 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..600

Cf. A128320 (triangle), A128322 (column 1), A128323 (column 2), A128324 (row sums); variant: A115081.

Cf. A000108 (Catalan numbers).

I:=[1,1,4]; [n le 3 select I[n] else (-(n-2)*(n-3)*Self(n-1) + 4*(3*(n-2)^2+n-3)*Self(n-2) + 8*(n-3)^2*(n-1)*Self(n-3))/n: n in [1..30]]; // G. C. Greubel, Jun 25 2024

a[n_]:= a[n]= If[n<3, (n!)^2, (-(n-1)*(n-2)*a[n-1] +4*(3*n^2-5*n +1)*a[n-2] + 8*(n-2)^2*n*a[n-3])/(n+1)];
Table[a[n], {n,0,40}] (* G. C. Greubel, Jun 25 2024 *)

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

@CachedFunction
def a(n): # a = A128321
    if n<3: return (1,1,4)[n]
    else: return (-(n-1)*(n-2)*a(n-1) + 4*(3*n^2-5*n+1)*a(n-2) + 8*(n-2)^2*n*a(n-3))/(n+1)
[a(n) for n in range(31)] # G. C. Greubel, Jun 25 2024

a(n) = Sum_{k=0..floor(n/2)} A000108(n-k)*A000108(k)*(k+1)!*C(n,2*k).
a(n) = Sum_{k=0..floor((n+1)/2)} ((k+1)!*C(2*(n-k), n-k)*C(2*k, k)*C(n, 2*k))/((k+1)*(n-k+1)).
a(n) = ( -(n-1)*(n-2)*a(n-1) + 4*(3*n^2 -5*n +1)*a(n-2) + 8*n*(n-2)^2* a(n-3) )/(n+1), with a(0) = 1, a(1) = 1, a(2) = 4. - G. C. Greubel, Jun 25 2024
a(n) ~ 2^(3*n/2 + 1) * exp(sqrt(2*n) - n/2 - 1/2) * n^((n-3)/2) / sqrt(Pi) * (1 - 7/(3*sqrt(2*n))). - Vaclav Kotesovec, Jun 25 2024

A128323 Column 2 of triangle A128320.

Original entry on oeis.org

1, 5, 12, 73, 418, 3110, 23920, 207549, 1885166, 18417710, 187881112, 2018628090, 22533601892, 261966343388, 3149344100224, 39158513865053, 501507474201750, 6611648592425790, 89492095211184360, 1242626064512513070

Offset: 0

Paul D. Hanna, Feb 25 2007

nonn

Cf. A128320 (triangle), A128321 (column 0), A128322 (column 1), A128324 (row sums); variant: A115083.

PARI

A128324 Row sums of triangle A128320.

Original entry on oeis.org

1, 2, 8, 31, 159, 955, 6677, 51308, 429868, 3847548, 36599474, 366515450, 3848812068, 42154442638, 480103999452, 5666303543327, 69139729751267, 870092119451903, 11272494698299169, 150074841511853609

Offset: 0

Paul D. Hanna, Feb 25 2007

nonn

Cf. A128320 (triangle), A128321 (column 0), A128322 (column 1), A128323 (column 2); variant: A115084.

PARI

cp's OEIS Frontend

A128320 Triangle, read by rows, where T(n,k) equals the dot product of the vector of terms in row n that are to the right of T(n,k) with the vector of terms in column k that are above T(n,k) for n>k+1>0, with the odd numbers in the secondary diagonal and all 1's in the main diagonal.

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A128321 Column 0 of triangle A128320.

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A128323 Column 2 of triangle A128320.

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A128324 Row sums of triangle A128320.

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