reviewed

approved

proposed

reviewed

editing

proposed

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, ...

proposed

editing

editing

proposed

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.

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

Row sums are A047055.

G. C. Greubel, <a href="/A142460/b142460.txt">Rows n = 1..50 of the triangle, flattened</a>

m=5; tT(n, k, m) = (m*n - m*k + 1)t*T(n - 1, k - 1, m) + (m*k - (m-1))t*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).

{1},

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},;

{1, 4031069, 622333256, 9346191344, 32208325226, 32208325226, 9346191344, 622333256, 4031069, 1}

T[n_, k_, m=5; ( Pascal level_]: k=N-1) A T[n_, 1, k, m] := If[k==1; A[n_, || k==n_] := , 1; A[n_, k_] := , (m*n - m*k + 1)A*T[n - 1, k - 1, m] + (m*k - (m - +1))A*T[n - 1, k]; a = Table[A[n, k, m], {n, 10}, {k, n} ]; Flatten[a]

Table[T[n, k, 5], {n, 1, 10}, {k, 1, n}]//Flatten (* modified by G. C. Greubel, Mar 14 2022 *)

(Sage)

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

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).

approved

editing

editing

approved

Row sums are {1, 2, 14, 168, 2856, 62832, 1696464, 54286848, 2008613376, 84361761792, ...}.

Row sums are A047055.

approved

editing

editing

approved

nonn,tabl,changedeasy