A188589 -id:A188589 - cp's OEIS frontend

A188587 1-Euler triangle.

1, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 14, 30, 14, 1, 1, 33, 146, 146, 33, 1, 1, 72, 603, 1168, 603, 72, 1, 1, 151, 2241, 7687, 7687, 2241, 151, 1, 1, 310, 7780, 44194, 76870, 44194, 7780, 310, 1, 1, 629, 25820, 231236, 649514, 649514, 231236, 25820, 629, 1, 1, 1268, 83121, 1131504, 4866222, 7794168, 4866222, 1131504, 83121, 1268, 1

Offset: 0

Paul Barry, Apr 04 2011

Formed with the same recurrence as the Euler triangle A008292 (adjusted for offset), but with the middle element of row n=2 set to 1.

Row sums are A188588. Second column is A188589.

			Triangle begins
  1;
  1,   1;
  1,   1,     1;
  1,   5,     5,      1;
  1,  14,    30,     14,      1;
  1,  33,   146,    146,     33,      1;
  1,  72,   603,   1168,    603,     72,      1;
  1, 151,  2241,   7687,   7687,   2241,    151,     1;
  1, 310,  7780,  44194,  76870,  44194,   7780,   310,   1;
  1, 629, 25820, 231236, 649514, 649514, 231236, 25820, 629, 1;

A188587 := proc(n,k) if k < 0 or k > n then 0; elif k=0 or k= n or n=2 then 1; else (n-k+1)*procname(n-1,k-1)+(k+1)*procname(n-1,k) ; end if; end proc:
seq(seq(A188587(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Apr 13 2011

T(n,k) = 0 if k < 0 or k > n.
T(n,k) = 1 if k=0 or k=n or n=2.
T(n,k)= (n-k+1)*T(n-1,k-1) + (k+1)*T(n-1,k), n > 2 and 1 <= k < n.

A336014 Irregular triangle read by rows: T(n,1) = T(n,2) = T(n,3n-2) = T(n,3n-1) = n for n >= 1 and T(n,k) = T(n-1,k-2) + T(n-1,k-1) for n > 1, 3 <= k <= 3*(n-1).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 3, 3, 4, 4, 6, 7, 8, 8, 8, 7, 6, 4, 4, 5, 5, 8, 10, 13, 15, 16, 16, 15, 13, 10, 8, 5, 5, 6, 6, 10, 13, 18, 23, 28, 31, 32, 31, 28, 23, 18, 13, 10, 6, 6, 7, 7, 12, 16, 23, 31, 41, 51, 59, 63, 63, 59, 51, 41, 31, 23, 16, 12, 7, 7

Offset: 1

Lechoslaw Ratajczak, Jul 04 2020

The number of terms in row n is 3*n-1 = A016789(n-1).
The sum of row n is equal to 2*A094002(n-1) = 2*A188589(n).
Fibonacci(n) = T(n+k,n) - T(n+k-1,n) for n >= 1, k = 1,2,3,...
The elements b(k) of the main diagonal, superdiagonal 1 and all subdiagonals have the recursive formula: b(k) = 2*b(k-1) + b(k-2) - 2*b(k-3) - b(k-4) for k > 4.

			Triangle begins:
n\k 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20...
1   1  1
2   2  2  2  2  2
3   3  3  4  4  4  4  3  3
4   4  4  6  7  8  8  8  7  6  4  4
5   5  5  8 10 13 15 16 16 15 13 10  8  5  5
6   6  6 10 13 18 23 28 31 32 31 28 23 18 13 10  6  6
7   7  7 12 16 23 31 41 51 59 63 63 59 51 41 31 23 16 12  7  7
...

Superdiagonal 1 is A029907 for n >= 1.
The main diagonal is A208354 for n >= 1.
Subdiagonal 1 is A102702(n-1) for n >= 1.
Subdiagonal 2 is A206268(n+2) for n >= 1 (conjectured).
Subdiagonal 3 is A191830(n+3) for n >= 1.
Cf. A007318, A051597, A228196.

T(n,k) = T(n,3*k-n) for 1 <= k <= 3*n-1.
T(n,k) = Sum_{u=2*(n-k)+3..2*n-k+1} ceiling(u/2)*A065941(k-2,u-2*(n-k)-3) for n >= 3, 3 <= k <= n.
T(n,k) = Sum_{m1=1..k-n} A208354(m1)*binomial(n-m1-1, k-n-m1) + Sum_{m2=1..2*n-k} A208354(m2)*binomial(n-m2-1, 2*n-k-m2) for n >= 2, n+1 <= k <= 2*n-1.
T(n,k) = Sum_{u=2*(k-2*n)+3..k-n+1} ceiling(u/2)*A065941(3*n-k-2,u-2*(k-2*n)-3) for n>= 3, 2*n <= k <= 3*(n-1).
T(n,k) = A208354(k) + (n-k)*Fibonacci(k) for n >= 3, 3 <= k <= n.
T(n,k) = A029907(k-1) + (n-k+1)*Fibonacci(k) for n >= 2, 3 <= k <= n+1.

cp's OEIS Frontend

A188587 1-Euler triangle.

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A336014 Irregular triangle read by rows: T(n,1) = T(n,2) = T(n,3n-2) = T(n,3n-1) = n for n >= 1 and T(n,k) = T(n-1,k-2) + T(n-1,k-1) for n > 1, 3 <= k <= 3*(n-1).

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cp's OEIS Frontend

A188587 1-Euler triangle.

Views

Author

Keywords

Comments

Examples

Programs

Maple

Formula

A336014 Irregular triangle read by rows: T(n,1) = T(n,2) = T(n,3*n-2) = T(n,3*n-1) = n for n >= 1 and T(n,k) = T(n-1,k-2) + T(n-1,k-1) for n > 1, 3 <= k <= 3*(n-1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A336014 Irregular triangle read by rows: T(n,1) = T(n,2) = T(n,3n-2) = T(n,3n-1) = n for n >= 1 and T(n,k) = T(n-1,k-2) + T(n-1,k-1) for n > 1, 3 <= k <= 3*(n-1).