A142458 -id:A142458 - cp's OEIS frontend

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

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

A155491 Triangle T(n, k) = binomial(n+1, k)*A142458(n+1, k+1)/(k+1), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 12, 1, 1, 78, 78, 1, 1, 415, 1820, 415, 1, 1, 2031, 27410, 27410, 2031, 1, 1, 9534, 330225, 959350, 330225, 9534, 1, 1, 43660, 3488884, 23935450, 23935450, 3488884, 43660, 1, 1, 196569, 33888576, 484631574, 1120179060, 484631574, 33888576, 196569, 1

Offset: 0

Roger L. Bagula, Jan 23 2009

			Triangle begins as:
  1;
  1,      1;
  1,     12,        1;
  1,     78,       78,         1;
  1,    415,     1820,       415,          1;
  1,   2031,    27410,     27410,       2031,         1;
  1,   9534,   330225,    959350,     330225,      9534,        1;
  1,  43660,  3488884,  23935450,   23935450,   3488884,    43660,      1;
  1, 196569, 33888576, 484631574, 1120179060, 484631574, 33888576, 196569, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A001263 (m=0), A155467 (m=1), this sequence (m=3), A155493 (m=4).

Cf. A142458.

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]];
T[n_, k_, m_]:= Binomial[n+1,k]*t[n+1,k+1,m]/(k+1);
Table[T[n,k,3], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Apr 01 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 T(n,k,m): return binomial(n+1,k)*t(n+1,k+1,m)/(k+1)
flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 01 2022

T(n, k) = binomial(n+1, k)*t(n, k, m)/(k+1), where t(n,k,m) = (m*(n-k)+1)*t(n-1,k-1,m) + (m*k-m+1)*t(n-1,k,m), t(n,1,m) = t(n,n,m) = 1, and m = 3.
From G. C. Greubel, Apr 01 2022: (Start)
T(n, k) = binomial(n+1, k)*A142458(n+1, k+1)/(k+1).
T(n, n-k) = T(n, k). (End)

Edited by G. C. Greubel, Apr 01 2022

A172010 a(n) = 2A142458(2n, n)/(n+1).

Original entry on oeis.org

1, 26, 2741, 683870, 315704418, 234725594388, 257237392999893, 390832857108454838, 787178784737043042806, 2031210797603911366282796, 6536955866068372922068141666, 25676217636579568989377656129516, 120915166829869713032692550819662756, 672580820552232143302651758669053327784

Offset: 1

Roger L. Bagula, Nov 19 2010

nonn

G. C. Greubel, Table of n, a(n) for n = 1..190

Cf. A142458, A177042, A177043.

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]];
a[n_]:= 2*T[2*n,n,3]/(n+1);
Table[a[n], {n,30}] (* modified by G. C. Greubel, Mar 14 2022 *)

@CachedFunction
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)
[2*T(2*n,n,3)/(n+1) for n in (1..30)] # G. C. Greubel, Mar 14 2022

a(n) = 2*A142458(2*n, n)/(n+1).

Name corrected and more terms added by G. C. Greubel, Mar 14 2022

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

A225433 Triangle T(n, k) = T(n, k-1) + (-1)^k*A142458(n+2, k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, -38, 1, 1, -165, -165, 1, 1, -676, 4806, -676, 1, 1, -2723, 44452, 44452, -2723, 1, 1, -10914, 362895, -1346780, 362895, -10914, 1, 1, -43681, 2780367, -20297327, -20297327, 2780367, -43681, 1, 1, -174752, 20554588, -263879264, 683233990, -263879264, 20554588, -174752, 1

Offset: 0

Roger L. Bagula, May 07 2013

			The triangle begins:
  1;
  1,      1;
  1,    -38,       1;
  1,   -165,    -165,         1;
  1,   -676,    4806,      -676,         1;
  1,  -2723,   44452,     44452,     -2723,       1;
  1, -10914,  362895,  -1346780,    362895,  -10914,      1;
  1, -43681, 2780367, -20297327, -20297327, 2780367, -43681, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A007318, A060187, A142458, A159041, A225356.

Maple
```
See Maple program in A159041.
```

(* First program *)
T[n_, k_, m_]:= T[n, k, m]= If[k==0 || k==n, 1, (m*(n+1) -m*(k+1) +1)*T[n-1, k- 1, m] + (m*(k+1) -(m-1))*T[n-1, k, m] ];
p[x_, n_]:= p[x, n]= Sum[x^i*If[i==Floor[n/2] && Mod[n, 2]==0, 0, If[i<= Floor[n/2], (-1)^i*T[n,i,3], -(-1)^(n-i)*T[n,i,3]]], {i,0,n}]/(1-x);
Flatten[Table[CoefficientList[p[x, n], x], {n,0,12}]]
(* Second program *)
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]];
A142458[n_, k_]:= T[n,k,3];
A225433[n_, k_]:= A225433[n, k]= If[k==0 || k==n, 1, If[k<=Floor[n/2], A225433[n, k-1] +(-1)^k*A142458[n+2, k+1], A225433[n, n-k]]];
Table[A225433[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 19 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 A142458(n,k): return T(n,k,3)
@CachedFunction
def A225433(n,k):
    if (k==0 or k==n): return 1
    elif (k <= (n//2)): return A225433(n,k-1) + (-1)^k*A142458(n+2,k+1)
    else: return A225433(n,n-k)
flatten([[A225433(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 19 2022

From G. C. Greubel, Mar 19 2022: (Start)
T(n, k) = T(n, k-1) + (-1)^k*A142458(n+2, k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1.
T(n, n-k) = T(n, k). (End)

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

Edited by G. C. Greubel, Mar 19 2022

A154425 a(n) = A142458(n, 1 + floor(n/2)).

Original entry on oeis.org

1, 1, 8, 39, 546, 5482, 109640, 1709675, 44451550, 947113254, 30307624128, 821539580358, 31218504053604, 1028949571999572, 45273781167981168, 1758747856988046771, 87937392849402338550, 3935893923685215214030

Offset: 1

Roger L. Bagula, Jan 09 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 1..375

Cf. A142458.

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]];
A154425[n_]:= T[n, 1+Floor[n/2], 3];
Table[A154425[n], {n, 30}] (* modified by G. C. Greubel, Mar 16 2022 *)

@CachedFunction
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)
def A154425(n): return T(n, 1 + (n//2), 3)
[A154425(n) for n in (1..30)] # G. C. Greubel, Mar 16 2022

a(n) = A142458(n, 1 + floor(n/2)).

Edited by G. C. Greubel, Mar 16 2022

A154596 a(n) = Sum_{j=1..n-1} A142458(n-1, k)*a(n - k), with a(1) = 1.

Original entry on oeis.org

1, 1, 2, 11, 129, 3214, 162491, 16306117, 3231430542, 1254563121783, 953359099059949, 1417753660258148022, 4128222097278496550683, 23571703478682225135264061, 264268834213603744830353397238

Offset: 1

Roger L. Bagula and Gary W. Adamson, Jan 12 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 1..80

Cf. A000670, A142458.

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]];
A142458[n_, k_]:= A142458[n, k] = T[n, k, 3];
a[n_]:= a[n]= If[n==1, 1, Sum[A142458[n-1, j]*a[n-j], {j,n-1}]];
Table[a[n], {n, 30}] (* modified by G. C. Greubel, Mar 16 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 A142458(n,k): return T(n,k,3)
@CachedFunction
def A154596(n): return 1 if (n==1) else sum( A142458(n-1, j)*A154596(n-j) for j in (1..n-1) )
[A154596(n) for n in (1..30)] # G. C. Greubel, Mar 16 2022

a(n) = Sum_{j=1..n-1} A142458(n-1, k)*a(n-k), with a(1) = 1.

Offset changed by G. C. Greubel, Mar 16 2022

A167786 Triangle of z Transform coefficients from General Pascal [1,8,1} A142458 polynomials multiplied by factor 3^Floor[(2*k - 1)/3].

Original entry on oeis.org

0, 3, 3, 6, 9, 45, 45, 27, 234, 540, 360, 27, 315, 1305, 1980, 990, 81, 1026, 6750, 18360, 20790, 8316, 243, 3807, 26379, 115830, 234630, 212058, 70686, 243, 7938, 37800, 177660, 582120, 939708, 706860, 201960, 729, 26001, 280827, 873180, 3087315

Offset: 0

Roger L. Bagula, Nov 12 2009

Row sums are:
{0, 3, 9, 99, 1161, 4617, 55323, 663633, 2654289, 31850739, 382206681...}
These are a sequence of Infinite sums that give A142458.
Even terms are factored by (1+2*n) which is the MacMahon (1+2*n)^k,but the polynomials seem fundamental
other than that.
A060187 MacMahon gives A013609 Triangle of coefficients in expansion of (1 + 2x)^n.
I looked for a simple infinite sum for the {1,8,1} and failed.
This reasoning comes from finding that the general z Transform polynomials are
related to the Eulerian: in fact this type of Eulerian polynomials A008292 gives (1+n)^k binomial.
The polynomials given here form a set of infinite sum sequences.

			{0},
{3},
{3, 6},
{9, 45, 45},
{27, 234, 540, 360},
{27, 315, 1305, 1980, 990},
{81, 1026, 6750, 18360, 20790, 8316},
{243, 3807, 26379, 115830, 234630, 212058, 70686},
{243, 7938, 37800, 177660, 582120, 939708, 706860, 201960},
{729, 26001, 280827, 873180, 3087315, 8058204, 10814958, 6967620, 1741905},
{2187, -308610, 1076490, 7334820, 17120565, 48411594, 104968710, 120570120, 67934295, 15096510}

A142458, A060187, A008292

m = 3 A[n_, 1] := 1 A[n_, n_] := 1
A[n_, k_] := (m*n - m*k + 1)A[n - 1, k - 1] + (m*k - (m - 1))A[n - 1, k]
a = Table[A[n, k], {n, 10}, {k, n}]
p[x_, n_] = x*Sum[a[[n, k]]*x^(k - 1), {k, 1, n}]/(x - 1)
b = Table[p[x, n], {n, 0, 10}]
Table[3^Floor[(2*k - 1)/3]*CoefficientList[ExpandAll[ InverseZTransform[b[[k]], x, n] /. UnitStep[ -1 + n] -> 1], n], {k, 1, Length[b]}]

m=3;
A(n,k)= (m*n - m*k + 1)A(n - 1, k - 1} + (m*k - (m - 1))A(n - 1, k)
q(n,k)=InverseZTransform[x*Sum[a[[n, k]]*x^(k - 1), {k, 1, n}]/(x - 1)^n, x, k]
out_n,k=3^Floor[(2*k - 1)/3]*coefficients(q[n,k])

A168295 Triangle T(n, k) = coefficients of (p(x,n)), where p(x, n) = (n-1)! * Sum_{j=1..n} A142458(n, j)*binomial(x+j-1, n-1), read by rows.

Original entry on oeis.org

1, 1, 2, 2, 10, 10, 6, 52, 120, 80, 24, 280, 1160, 1760, 880, 120, 1520, 10000, 27200, 30800, 12320, 720, 11280, 78160, 343200, 695200, 628320, 209440, 5040, 164640, 784000, 3684800, 12073600, 19490240, 14660800, 4188800, 40320, 1438080, 15532160, 48294400, 170755200, 445688320, 598160640, 385369600, 96342400

Offset: 1

Roger L. Bagula, Nov 22 2009

			Triangle begins as:
     1;
     1,      2;
     2,     10,     10;
     6,     52,    120,      80;
    24,    280,   1160,    1760,      880;
   120,   1520,  10000,   27200,    30800,    12320;
   720,  11280,  78160,  343200,   695200,   628320,   209440;
  5040, 164640, 784000, 3684800, 12073600, 19490240, 14660800, 4188800;

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

Cf. A008544, A142458, A167786.

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]];
A142458[n_, k_]:= A142458[n, k]= T[n,k,3];
p[x_, n_]:= p[x, n]= Sum[A142458[n,k]*Pochhammer[x+k-n+1, n-1], {k, n}];
Table[CoefficientList[p[x, n], x], {n, 1, 12}]//Flatten (* modified by G. C. Greubel, Mar 17 2022 *)

@CachedFunction
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)
def A142458(n,k): return T(n,k,3)
@CachedFunction
def p(n,x): return sum( A142458(n,j)*rising_factorial(x+j-n+1, n-1) for j in (1..n))
def A168295(n,k): return ( p(n,x) ).series(x,n+1).list()[k-1]
flatten([[ A168295(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 17 2022

T(n, k) = coefficients of (p(x,n)), where p(x, n) = (n-1)! * Sum_{j=1..n} A142458(n, j)*binomial(x+j-1, n-1).
From G. C. Greubel, Mar 17 2022: (Start)
T(n, k) = coefficients of (p(n, x)), where p(n, x) = Sum_{j=1..n} A142458(n, j)*Pochhammer(x+j-n+1, n-1).
T(n, 1) = (n-1)!.
T(n, n) = A008544(n-1). (End)

Edited by G. C. Greubel, Mar 17 2022

A171274 Matrix inverse of A142458.

Original entry on oeis.org

1, -1, 1, 7, -8, 1, -235, 273, -39, 1, 35353, -41116, 5928, -166, 1, -22683409, 26382125, -3804940, 106900, -677, 1, 60147266239, -69954818244, 10089231945, -283474190, 1796973, -2724, 1, -648088191536203, 753764796604717, -108711714513099, 3054442698125, -19362601277, 29358651, -10915, 1

Offset: 1

Roger L. Bagula and Mats Granvik, Dec 06 2009

			The triangle starts as:
            1;
           -1,            1;
            7,           -8,           1;
         -235,          273,         -39,          1;
        35353,       -41116,        5928,       -166,       1;
    -22683409,     26382125,    -3804940,     106900,    -677,     1;
  60147266239, -69954818244, 10089231945, -283474190, 1796973, -2724, 1;

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

Cf. A142458.

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:
A171274 := proc(n,k) option remember; if k=n then 1; else -add( procname(n,j)*A142458(j,k),j=k+1..n);  end if; end proc:
seq(seq(A171274(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]];
A142458[n_, k_]:= T[n,k,3];
A171274[n_, k_]:= A171274[n, k]= If[k==n, 1, -Sum[A171274[n, j]*A142458[j, k], {j,k+1,n}] ];
Table[A171274[n, k], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 18 2022 *)

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 A142458(n,k): return T(n,k,3)
@CachedFunction
def A171274(n,k):
    if (k==n): return 1
    else: return (-1)*sum( A171274(n,j)*A142458(j,k) for j in (k+1..n) )
flatten([[A171274(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 18 2022

Sum_{j=k..n} T(n,j)*A142458(j,k) = delta(n,k), the Kronecker delta.
T(n, k) = (-1)*Sum_{j=k+1..n} T(n, j)*A142458(j, k), with T(n, n) = 1. - R. J. Mathar, Jun 04 2011
From G. C. Greubel, Mar 18 2022: (Start)
Sum_{k=1..n} T(n, k) = 0^(n-1).
T(n, n-1) = (-1)*A142458(n, 2). (End)

cp's OEIS Frontend

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

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A155491 Triangle T(n, k) = binomial(n+1, k)*A142458(n+1, k+1)/(k+1), read by rows.

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A172010 a(n) = 2*A142458(2*n, n)/(n+1).

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

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A225433 Triangle T(n, k) = T(n, k-1) + (-1)^k*A142458(n+2, k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1, read by rows.

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A154425 a(n) = A142458(n, 1 + floor(n/2)).

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A154596 a(n) = Sum_{j=1..n-1} A142458(n-1, k)*a(n - k), with a(1) = 1.

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A172010 a(n) = 2A142458(2n, n)/(n+1).