A142560 -id:A142560 - cp's OEIS frontend

A142458 Triangle T(n,k) read by rows: T(n,k) = 1 if k=1 or k=n, otherwise T(n,k) = (3n-3k+1)T(n-1,k-1) + (3k-2)*T(n-1,k).

1, 1, 1, 1, 8, 1, 1, 39, 39, 1, 1, 166, 546, 166, 1, 1, 677, 5482, 5482, 677, 1, 1, 2724, 47175, 109640, 47175, 2724, 1, 1, 10915, 373809, 1709675, 1709675, 373809, 10915, 1, 1, 43682, 2824048, 23077694, 44451550, 23077694, 2824048, 43682, 1

Offset: 1

Roger L. Bagula, Sep 19 2008

Consider the triangle T(n,k) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k). For m = ...,-2,-1,0,1,2,3,... we get ..., A225372, A144431, A007318, A008292, A060187, A142458, ... - N. J. A. Sloane, May 08 2013

			The rows n >= 1 and columns 1 <= k <= n look as follows:
  1;
  1,     1;
  1,     8,       1;
  1,    39,      39,        1;
  1,   166,     546,      166,        1;
  1,   677,    5482,     5482,      677,        1;
  1,  2724,   47175,   109640,    47175,     2724,       1;
  1, 10915,  373809,  1709675,  1709675,   373809,   10915,     1;
  1, 43682, 2824048, 23077694, 44451550, 23077694, 2824048, 43682, 1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
G. Strasser, Generalisation of the Euler adic, Math. Proc. Camb. Phil. Soc. 150 (2010) 241-256, Triangle A_3(n,k).

Cf. A225372 (m=-2), A144431 (m=-1), A007318 (m=0), A008292 (m=1), A060187 (m=2), this sequence (m=3), A142459 (m=4), A142560 (m=5), A142561 (m=6), A142562 (m=7), A167884 (m=8), A257608 (m=9).

Cf. A008544, A142976, A144380, A144381, A144414.

A142458 := proc(n,k) if n = k then 1; elif k > n or k < 1 then 0 ;else (3*n-3*k+1)*procname(n-1,k-1)+(3*k-2)*procname(n-1,k) ; end if; end proc:
seq(seq(A142458(n,k),k=1..n),n=1..10) ; # R. J. Mathar, Jun 04 2011

T[n_, k_, m_]:= T[n, k, m]= If[k==1 || k==n, 1, (m*n-m*k+1)*T[n-1, k-1, m] + (m*k -m+1)*T[n-1, k, m] ];
Table[T[n, k, 3], {n, 1, 10}, {k, 1, n}]//Flatten (* modified by G. C. Greubel, Mar 14 2022 *)

def T(n,k,m): # A142458
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m)
flatten([[T(n,k,3) for k in (1..n)] for n in (1..10)]) # G. C. Greubel, Mar 14 2022

T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), with T(n, 1) = T(n, n) = 1, and m = 3.
Sum_{k=1..n} T(n, k) = A008544(n-1).
From G. C. Greubel, Mar 14 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 2) = A144414(n-1).
T(n, 3) = A142976(n-2).
T(n, 4) = A144380(n-3).
T(n, 5) = A144381(n-4). (End)

Edited by the Associate Editors of the OEIS, Aug 28 2009

A142460 Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (mn-mk+1)T(n-1,k-1) + (mk-m+1)*T(n-1,k), where m = 5.

Original entry on oeis.org

1, 1, 1, 1, 12, 1, 1, 83, 83, 1, 1, 514, 1826, 514, 1, 1, 3105, 28310, 28310, 3105, 1, 1, 18656, 376615, 905920, 376615, 18656, 1, 1, 111967, 4627821, 22403635, 22403635, 4627821, 111967, 1, 1, 671838, 54377008, 478781506, 940952670, 478781506, 54377008, 671838, 1

Offset: 1

Roger L. Bagula, Sep 19 2008

One of a family of triangles. For m = ...,-2,-1,0,1,2,3,4,5,... we get ..., A225372, A144431, A007318, A008292, A060187, A142458, A142459, A142560, ...

			Triangle begins as:
  1;
  1,      1;
  1,     12,        1;
  1,     83,       83,         1;
  1,    514,     1826,       514,         1;
  1,   3105,    28310,     28310,      3105,         1;
  1,  18656,   376615,    905920,    376615,     18656,        1;
  1, 111967,  4627821,  22403635,  22403635,   4627821,   111967,      1;
  1, 671838, 54377008, 478781506, 940952670, 478781506, 54377008, 671838, 1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
G. Strasser, Generalisation of the Euler adic, Math. Proc. Camb. Phil. Soc. 150 (2010) 241-256, Triangle A_5(n,k).

Cf. A225372 (m=-2), A144431 (m=-1), A007318 (m=0), A008292 (m=1), A060187 (m=2), A142458 (m=3), A142459 (m=4), this sequence (m=5), A142561 (m=6), A142562 (m=7), A167884 (m=8), A257608 (m=9).

Cf. A047055 (row sums).

A142460 := proc(n, k) if n = k then 1; elif k > n or k < 1 then 0 ; else (5*n-5*k+1)*procname(n-1, k-1)+(5*k-4)*procname(n-1, k) ; end if; end proc:
seq(seq(A142459(n, k), k=1..n), n=1..10) ; # R. J. Mathar, May 11 2013

T[n_, k_, m_]:= T[n, k, m]= If[k==1 || k==n, 1, (m*n-m*k+1)*T[n-1, k-1, m] + (m*k -m+1)*T[n-1, k, m] ];
Table[T[n, k, 5], {n, 1, 10}, {k, 1, n}]//Flatten (* modified by G. C. Greubel, Mar 14 2022 *)

def T(n,k,m): # A142460
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m)
flatten([[T(n,k,5) for k in (1..n)] for n in (1..10)]) # G. C. Greubel, Mar 14 2022

T(n, k, m) = (m*n - m*k + 1)*T(n-1, k-1, m) + (m*k - (m-1))*T(n-1, k, m), with T(t,1,m) = T(n,n,m) = 1, and m = 5.

Sum_{k=1..n} T(n, k, 5) = A047055(n-1).

Edited by N. J. A. Sloane, May 08 2013, May 11 2013

A225372 Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (mn-mk+1)T(n-1,k-1) + (mk-m+1)*T(n-1,k), where m = -2.

Original entry on oeis.org

1, 1, 1, 1, -2, 1, 1, -1, -1, 1, 1, -4, 6, -4, 1, 1, -3, 2, 2, -3, 1, 1, -6, 15, -20, 15, -6, 1, 1, -5, 9, -5, -5, 9, -5, 1, 1, -8, 28, -56, 70, -56, 28, -8, 1, 1, -7, 20, -28, 14, 14, -28, 20, -7, 1, 1, -10, 45, -120, 210, -252, 210, -120, 45, -10, 1

Offset: 1

N. J. A. Sloane and Roger L. Bagula, May 08 2013

			Triangle begins:
  1;
  1,  1;
  1, -2,  1;
  1, -1, -1,   1;
  1, -4,  6,  -4,  1;
  1, -3,  2,   2, -3,   1;
  1, -6, 15, -20, 15,  -6,   1;
  1, -5,  9,  -5, -5,   9,  -5,  1;
  1, -8, 28, -56, 70, -56,  28, -8,  1;
  1, -7, 20, -28, 14,  14, -28, 20, -7, 1;

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

For m = ...,-2,-1,0,1,2,3,4,5,6,7,8, ... we get ..., A225372, A144431, A007318, A008292, A060187, A142458, A142459, A142560, A142561, A142562, A167884, ...

Cf. A130706 (row sums).

function T(n,k,m)
  if k eq 1 or k eq n then return 1;
  else return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m);
  end if; return T;
end function;
A225372:= func< n,k | T(n,k,-2) >;
[A225372(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 17 2022

T:=proc(n,k,l) option remember;
if (n=1 or k=1 or k=n) then 1 else
(l*n-l*k+1)*T(n-1,k-1,l)+(l*k-l+1)*T(n-1,k,l); fi; end;
for n from 1 to 14 do lprint([seq(T(n,k,-2),k=1..n)]); od;

T[n_, k_, l_] := T[n, k, l] = If[n == 1 || k == 1 || k == n, 1, (l*n-l*k+1)*T[n-1, k-1, l]+(l*k-l+1)*T[n-1, k, l]]; Table[T[n, k, -2], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 09 2014, translated from Maple *)

@CachedFunction
def T(n,k,m):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m)
def A225372(n,k): return T(n,k,-2)
flatten([[ A225372(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 17 2022

T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), with T(n, 1) = T(n, n) = 1, and m = -2.

Sum_{k=1..n} T(n, k) = A130706(n-1). - G. C. Greubel, Mar 17 2022

cp's OEIS Frontend

A142458 Triangle T(n,k) read by rows: T(n,k) = 1 if k=1 or k=n, otherwise T(n,k) = (3*n-3*k+1)*T(n-1,k-1) + (3*k-2)*T(n-1,k).

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A142460 Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 5.

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A225372 Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = -2.

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Magma

Maple

Mathematica

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A142458 Triangle T(n,k) read by rows: T(n,k) = 1 if k=1 or k=n, otherwise T(n,k) = (3n-3k+1)T(n-1,k-1) + (3k-2)*T(n-1,k).

A142460 Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (mn-mk+1)T(n-1,k-1) + (mk-m+1)*T(n-1,k), where m = 5.

A225372 Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (mn-mk+1)T(n-1,k-1) + (mk-m+1)*T(n-1,k), where m = -2.