A142976 -id:A142976 - 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

A144380 Third subdiagonal of A142458: a(n) = A142458(n+3,n).

Original entry on oeis.org

1, 166, 5482, 109640, 1709675, 23077694, 284433852, 3300384000, 36740695125, 397251942790, 4206505251886, 43874389439176, 452588032465727, 4630933106076350, 47101176806668160, 476947462419456864, 4813761757416769257, 48466731584985480870, 487104579690137249650, 4889039701269534580360

Offset: 1

Roger L. Bagula and Gary W. Adamson, Oct 01 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 1..990
Index entries for linear recurrences with constant coefficients, signature (40,-675,6294,-35679,127548,-289173,409062,-347112,161056,-31360).

Cf. A142458, A142976.

[(1/162)*( 8*10^(n+3) - 30*(3*n +8)*7^(n+2) + 6*(9*n^2 +39*n +40)*4^(n+2) - (27*n^3 +135*n^2 +198*n +80)): n in [1..30]]; // G. C. Greubel, Mar 15 2022

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]];
A144380[n_]:= T[n+3, n, 3];
Table[A144380[n], {n,30}] (* modified by G. C. Greubel, Mar 15 2022 *)

@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 A144380(n): return T(n+3, n, 3)
[A144380(n) for n in (1..30)] # G. C. Greubel, Mar 15 2022

G.f.: x*(1 +126*x -483*x^2 -3884*x^3 +15300*x^4 -10848*x^5 -8960*x^6)/ ( (1-10*x) *(1-7*x)^2 *(1-4*x)^3 *(1-x)^4 ). - R. J. Mathar, Sep 14 2013

a(n) = (1/162)*( 8*10^(n+3) - 30*(3*n +8)*7^(n+2) + 6*(9*n^2 +39*n +40)*4^(n+2) - (27*n^3 +135*n^2 +198*n +80)). - G. C. Greubel, Mar 15 2022

A144414 a(n) = 2*(4^n - 1)/3 - n.

Original entry on oeis.org

1, 8, 39, 166, 677, 2724, 10915, 43682, 174753, 699040, 2796191, 11184798, 44739229, 178956956, 715827867, 2863311514, 11453246105, 45812984472, 183251937943, 733007751830, 2932031007381, 11728124029588, 46912496118419

Offset: 1

Roger L. Bagula and Gary W. Adamson, Oct 01 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-9,4).

Cf. A142458, A142976, A144380.

[(2^(2*n+1) -3*n -2)/3: n in [1..50]]; // G. C. Greubel, Mar 28 2021

Table[2(4^n-1)/3 -n,{n,30}] (* or *) LinearRecurrence[{6,-9,4},{1,8,39},30] (* Harvey P. Dale, Mar 17 2015 *)

[(2^(2*n+1) -3*n -2)/3 for n in (1..50)] # G. C. Greubel, Mar 28 2021

a(n) = A142458(n+1,n).
a(n) = A020988(n) - n. - R. J. Mathar, Nov 21 2008
G.f.: x*(1+2*x)/((1-x)^2*(1-4*x)). - Colin Barker, Jan 11 2012
a(1)=1, a(2)=8, a(3)=39, a(n) = 6*a(n-1) - 9*a(n-2) + 4*a(n-3). - Harvey P. Dale, Mar 17 2015
E.g.f.: (1/3)*(-2 - 3*x + 2*exp(x))*exp(x). - G. C. Greubel, Mar 28 2021

A144381 a(n) = A142458(n+5, n). Fifth diagonal of A142458 triangle.

Original entry on oeis.org

1, 677, 47175, 1709675, 44451550, 947113254, 17716715490, 302925749370, 4856552119935, 74258231957275, 1095758678253041, 15736592058221517, 221321453958111620, 3062416225698505060, 41836761536767296660, 565817483249269872324, 7591501608353930033805, 101209790951020335444705

Offset: 1

Roger L. Bagula and Gary W. Adamson, Oct 01 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 1..890
Index entries for linear recurrences with constant coefficients, signature (75, -2520, 50204, -661458, 6086718, -40273648, 194372208, -687083013, 1771618303, -3293261472, 4325310828, -3886563008, 2261691264, -765434880, 114150400).

Cf. A142458, A142976, A144380, A144414.

[(1/1944)*((27*n^3 + 216*n^2 + 549*n + 440)*(3*n + 2 - 2*4^(n + 4)) +
   60*(9*n^2 + 57*n + 88)*7^(n + 3) - 32*(3*n + 11)*10^(n + 4) +
880*13^(n + 3)): n in [1..30]]; // G. C. Greubel, Mar 16 2022

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]];
A144381[n_]:= T[n+4,n,3];
Table[A144381[n], {n,30}] (* modified by G. C. Greubel, Mar 16 2022 *)

@CachedFunction
def T(n,k,m): # A144381
    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 A144381(n): return T(n+4, n, 3)
[A144381(n) for n in (1..30)] # G. C. Greubel, Mar 16 2022

a(n) = A142458(n+5, n).
From G. C. Greubel, Mar 16 2022: (Start)
G.f.: x*(1 +602*x -1080*x^2 -172614*x^3 +1780275*x^4 -5025348*x^5 -7549548*x^6 +60043488*x^7 -99645984*x^8 +39979520*x^9 +27596800*x^10)/((1-x)^5*(1-4*x)^4*(1-7*x)^3*(1-10*x)^2*(1-13*x)).
a(n) = (1/1944)*((27*n^3 +216*n^2 +549*n +440)*(3*n +2 - 2*4^(n+4)) +
   60*(9*n^2 +57*n +88)*7^(n+3) -32*(3*n+11)*10^(n+4) + 880*13^(n+3)). (End)

Edited by G. C. Greubel, Mar 16 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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A144380 Third subdiagonal of A142458: a(n) = A142458(n+3,n).

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A144414 a(n) = 2*(4^n - 1)/3 - n.

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A144381 a(n) = A142458(n+5, n). Fifth diagonal of A142458 triangle.

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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).