A006008 -id:A006008 - cp's OEIS frontend

A144680 Triangle read by rows, lower half of an array formed by A004736 * A144328 (transform).

1, 2, 3, 3, 5, 7, 4, 7, 11, 14, 5, 9, 15, 21, 25, 6, 11, 19, 28, 36, 41, 7, 13, 23, 35, 47, 57, 63, 8, 15, 27, 42, 58, 73, 85, 92, 9, 17, 31, 49, 69, 89, 107, 121, 129, 10, 19, 35, 56, 80, 105, 129, 150, 166, 175

Offset: 1

Gary W. Adamson, Sep 19 2008

Triangle read by rows, lower half of an array formed by A004736 * A144328 (transform).

			The array is formed by A004736 * A144328 (transform) where A004736 = the natural number decrescendo triangle and A144328 = a crescendo triangle. First few rows of the array =
  1, 1,  1,  1,  1,  1, ...
  2, 3,  3,  3,  3,  3, ...
  3, 5,  7,  7,  7,  7, ...
  4, 7, 11, 14, 14, 14, ...
  5, 9, 15, 21, 25, 25, ...
  ...
Triangle begins as:
   1;
   2,  3;
   3,  5,  7;
   4,  7, 11, 14;
   5,  9, 15, 21, 25;
   6, 11, 19, 28, 36,  41;
   7, 13, 23, 35, 47,  57,  63;
   8, 15, 27, 42, 58,  73,  85,  92;
   9, 17, 31, 49, 69,  89, 107, 121, 129;
  10, 19, 35, 56, 80, 105, 129, 150, 166, 175;
  ...

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A004006, A006008, A027965, A050407, A074742, A144328.

T[n_, k_]:= (3*(k^2-k+2)*n - k*(k-1)*(2*k-1))/6;
Table[T[n, k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Oct 18 2021 *)

def A144680(n,k): return (3*(k^2-k+2)*n - k*(k-1)*(2*k-1))/6
flatten([[A144680(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Oct 18 2021

Sum_{k=1..n} T(n, k) = A006008(n).
From G. C. Greubel, Oct 18 2021: (Start)
T(n, k) = (1/6)*( 3*(k^2 - k + 2)*n - k*(k-1)*(2*k-1) ).
T(n, n) = A004006(n).
T(n, n-1) = A050407(n+2).
T(n, n-2) = A027965(n-1) = A074742(n-2). (End)

A337991 Triangle read by rows: T(n,m) = Sum_{i=1..n} C(n,i-m)C(n+m-i,i-1)C(n+m-i,m)/n, with T(0,0)=1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 5, 4, 1, 4, 13, 15, 7, 1, 9, 35, 52, 36, 11, 1, 21, 96, 175, 160, 75, 16, 1, 51, 267, 576, 655, 415, 141, 22, 1, 127, 750, 1869, 2541, 2030, 952, 245, 29, 1, 323, 2123, 6000, 9492, 9156, 5488, 1988, 400, 37, 1, 835, 6046, 19107, 34476, 38976, 28476, 13356, 3852, 621, 46, 1

Offset: 0

Vladimir Kruchinin, Oct 06 2020

			Triangle begins as:
   1;
   1,   1;
   1,   2,   1;
   2,   5,   4,   1;
   4,  13,  15,   7,   1;
   9,  35,  52,  36,  11,   1;
  21,  96, 175, 160,  75,  16,  1;
  51, 267, 576, 655, 415, 141, 22,  1;
  ...

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A001006, A005773, A086246, A132894.
Diagonals include: A000124, A006008.
Sums include: A000007 (signed row), A019590 (signed diagonal), A025227 (row), A102407 (diagonal).

B:=Binomial;
A337991:= func< n,k | n eq 0 select 1 else (1/n)*(&+[B(n, j-k)*B(n+k-j, j-1)*B(n+k-j, k): j in [1..n]]) >;
[A337991(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 31 2024

T[0, 0] = 1; T[n_, m_] := Sum[Binomial[n, i - m] * Binomial[n + m - i, i - 1] * Binomial[n + m - i, m]/n, {i, 1, n}]; Table[T[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Amiram Eldar, Oct 06 2020 *)

T(n,m):=if m=n then 1 else if n=0 then 0 else sum(binomial(n,i-m)*binomial(n+m-i,i-1)*binomial(n+m-i,m),i,1,n)/n;

def A337991(n,k):
    b=binomial
    if n==0: return 1
    else: return (1/n)*sum(b(n, j-k)*b(n+k-j, j-1)*b(n+k-j, k) for j in range(1,n+1))
# SageMath
flatten([[A337991(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 31 2024

G.f.: ( 1 - x*(y-1)- sqrt(x^2*(y^2-2*y-3) - 2*x*(y+1) + 1) )/(2*x).
From G. C. Greubel, Oct 31 2024: (Start)
T(n, k) = binomial(n, 1-k)*binomial(n+k-1, k)*Hypergeometric3F2([1-n, (1 -n -k)/2, (2-n-k)/2], [2-k, 1-n-k], 4), with T(0, 0) = 1.
T(n, 0) = A086246(n+1).
T(n, n-1) = A000124(n-1), n >= 1.
T(n, n-2) = A006008(n-1), n >= 2.
T(n, n-3) = (1/72)*(n^4 -6*n^3 +47*n^2 -114*n +144)*binomial(n-1,2), n >= 3.
T(n, n-4) = (1/480)*(n-2)*(n^4 -8*n^3 +99*n^2 -332*n +960)*binomial(n-1,3), n >= 4.
Sum_{k=0..n} T(n, k) = A025227(n+1).
Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A102407(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A019590(n+1). (End)

cp's OEIS Frontend

A144680 Triangle read by rows, lower half of an array formed by A004736 * A144328 (transform).

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A337991 Triangle read by rows: T(n,m) = Sum_{i=1..n} C(n,i-m)C(n+m-i,i-1)C(n+m-i,m)/n, with T(0,0)=1.

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cp's OEIS Frontend

A144680 Triangle read by rows, lower half of an array formed by A004736 * A144328 (transform).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

A337991 Triangle read by rows: T(n,m) = Sum_{i=1..n} C(n,i-m)*C(n+m-i,i-1)*C(n+m-i,m)/n, with T(0,0)=1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

Python

Formula

A337991 Triangle read by rows: T(n,m) = Sum_{i=1..n} C(n,i-m)C(n+m-i,i-1)C(n+m-i,m)/n, with T(0,0)=1.