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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A062145 Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.

Original entry on oeis.org

1, 1, 4, 1, 10, 10, 1, 18, 45, 20, 1, 28, 126, 140, 35, 1, 40, 280, 560, 350, 56, 1, 54, 540, 1680, 1890, 756, 84, 1, 70, 945, 4200, 7350, 5292, 1470, 120, 1, 88, 1540, 9240, 23100, 25872, 12936, 2640, 165, 1, 108, 2376, 18480, 62370, 99792, 77616, 28512, 4455, 220
Offset: 0

Views

Author

Wolfdieter Lang, Jun 19 2001

Keywords

Comments

Coefficient triangle of certain polynomials N(3; m,x).

Examples

			As a square array:
    1,    1,     1,     1,     1,     1,    1,  1, ... A000012;
    4,   10,    18,    28,    40,    54,   70, 88, ... A028552;
   10,   45,   126,   280,   540,   945, 1540, ....... A105938;
   20,  140,   560,  1680,  4200,  9240, ............. A105939;
   35,  350,  1890,  7350, 23100, 62370, ............. A027803;
   56,  756,  5292, 25872, 99792, .................... A105940;
   84, 1470, 12936, 77616, ........................... A105942;
  120, 2640, 28512, .................................. A105943;
  165, 4455, 57015, .................................. A105944;
  ....;
As a triangle:
  1;
  1,   4;
  1,  10,   10;
  1,  18,   45,    20;
  1,  28,  126,   140,    35;
  1,  40,  280,   560,   350,    56;
  1,  54,  540,  1680,  1890,   756,    84;
  1,  70,  945,  4200,  7350,  5292,  1470,   120;
  1,  88, 1540,  9240, 23100, 25872, 12936,  2640,  165;
  1, 108, 2376, 18480, 62370, 99792, 77616, 28512, 4455, 220;
  ....;

Links

Crossrefs

Family of polynomials: A008459 (c=1), A132813 (c=2), A062196 (c=3), this sequence (c=4), A062264 (c=5), A062190 (c=6).
Columns: A028552 (k=1), A105938 (k=2), A105939 (k=3), A027803 (k=4), A105940 (k=5), A105942 (k=6), A105943 (k=7), A105944 (k=8).
Diagonals: A000292 (k=n), A027800 (k=n-1), A107417 (k=n-2), A107418 (k=n-3), A107419 (k=n-4), A107420 (k=n-5), A107421 (k=n-6), A107422 (k=n-7).
Sums: A002054 (row).
Cf. A000108, A000292, A047074.

Programs

  • Magma
    A062145:= func< n,k | Binomial(n,k)*Binomial(n+3,k) >;
[A062145(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 07 2025
  • Mathematica
    NN[3, m_, x_] := x^m*(2*m+3)!*Hypergeometric2F1[-m, -m, -2*m-3, (x-1)/x]/( (m+3)!*m!); Table[CoefficientList[NN[3, m, x], x], {m, 0, 9}] // Flatten (* Jean-François Alcover, Sep 18 2013 *)
P[c_, n_, z_] := Sum[Binomial[n, k] Pochhammer[n-k+c, k] z^k /k!, {k,0,n}];
CL[c_] := Table[CoefficientList[P[c, n, z], z], {n, 0, 5}] // TableForm
CL[4]  (* Peter Luschny, Feb 12 2024 *)
A062145[n_,k_]:= Binomial[n,k]*Binomial[n+3,k];
Table[A062145[n,k], {n,0,12},{k,0,n}]//Flatten (* G. C. Greubel, Mar 07 2025 *)
  • SageMath
    def A062145(n,k): return binomial(n,k)*binomial(n+3,k)
print(flatten([[A062145(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Mar 07 2025

Formula

The e.g.f. of the m-th (unsigned) column sequence without leading zeros of the generalized (a=3) Laguerre triangle L(3; n+m, m) = A062137(n+m, m), n >= 0, is N(3; m, x)/(1-x)^(2*(m+2)), with the row polynomials N(3; m, x) := Sum_{k=0..m} a(m, k)*x^k.
N(3; m, x) := ((1-x)^(2*(m+2)))*(d^m/dx^m)(x^m/(m!*(1-x)^(m+4))); a(m, k) = [x^k]N(3; m, x).
N(3; m, x) = Sum_{j=0..m} ((binomial(m, j)*(2*m+3-j)!/((m+3)!*(m-j)!))*(x^(m-j))*(1-x)^j).
N(3; m, x)= x^m*(2*m+3)! * 2F1(-m, -m; -2*m-3; (x-1)/x)/((m+3)!*m!). - Jean-François Alcover, Sep 18 2013
From G. C. Greubel, Mar 07 2025 : (Start)
T(n, k) = binomial(n, k)*binomial(n+3, k).
T(2*n, n) = (1/2)*(n+1)^2*A000108(n)*A000108(n+2).
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^floor((n+2)/2)*(A047074(n+3) - A047074(n+ 2)). (End)

Extensions

New name by Peter Luschny, Feb 12 2024
More terms from G. C. Greubel, Mar 07 2025

A047072 Array A read by diagonals: A(h,k)=number of paths consisting of steps from (0,0) to (h,k) such that each step has length 1 directed up or right and no step touches the line y=x unless x=0 or x=h.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 3, 2, 2, 3, 1, 1, 4, 5, 4, 5, 4, 1, 1, 5, 9, 5, 5, 9, 5, 1, 1, 6, 14, 14, 10, 14, 14, 6, 1, 1, 7, 20, 28, 14, 14, 28, 20, 7, 1, 1, 8, 27, 48, 42, 28, 42, 48, 27, 8, 1, 1, 9, 35, 75, 90, 42, 42, 90, 75, 35, 9, 1
Offset: 0

Views

Author

Clark Kimberling

Keywords

Examples

			Array, A(n, k), begins as:
  1, 1,  1,  1,  1,   1,   1,   1, ...;
  1, 2,  1,  2,  3,   4,   5,   6, ...;
  1, 1,  2,  2,  5,   9,  14,  20, ...;
  1, 2,  2,  4,  5,  14,  28,  48, ...;
  1, 3,  5,  5, 10,  14,  42,  90, ...;
  1, 4,  9, 14, 14,  28,  42, 132, ...;
  1, 5, 14, 28, 42,  42,  84, 132, ...;
  1, 6, 20, 48, 90, 132, 132, 264, ...;
Antidiagonals, T(n, k), begins as:
  1;
  1,  1;
  1,  2,  1;
  1,  1,  1,  1;
  1,  2,  2,  2,  1;
  1,  3,  2,  2,  3,  1;
  1,  4,  5,  4,  5,  4,  1;
  1,  5,  9,  5,  5,  9,  5,  1;
  1,  6, 14, 14, 10, 14, 14,  6,  1;

Links

Crossrefs

Cf. A000007, A000027, A000096, A002420.
Cf. A005586, A025174, A047079, A063886.
Cf. A047073, A047074, A047079.
The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Diagonals give A000108, A000245, A002057, A000344, A003517, A000588, A003518, A003519, A001392, ...

Programs

  • Magma
    b:= func< n | n eq 0 select 1 else 2*Catalan(n-1) >;
function A(n,k)
  if k eq n then return b(n);
  elif k gt n then return Binomial(n+k-1, n) - Binomial(n+k-1, n-1);
  else return Binomial(n+k-1, k) - Binomial(n+k-1, k-1);
  end if; return A;
end function;
// [[A(n,k): k in [0..12]]: n in [0..12]];
T:= func< n,k | A(n-k, k) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 13 2022
  • Mathematica
    A[, 0]= 1; A[0, ]= 1; A[h_, k_]:= A[h, k]= If[(k-1>h || k-1Jean-François Alcover, Mar 06 2019 *)
  • SageMath
    def A(n,k):
    if (k==n): return 2*catalan_number(n-1) + 2*int(n==0)
    elif (k>n): return binomial(n+k-1, n) - binomial(n+k-1, n-1)
    else: return binomial(n+k-1, k) - binomial(n+k-1, k-1)
def T(n,k): return A(n-k, k)
# [[A(n,k) for k in range(12)] for n in range(12)]
flatten([[T(n,k) for k in range(n+1)] for n in range(12)]) # G. C. Greubel, Oct 13 2022

Formula

A(n, n) = 2*[n=0] - A002420(n),
A(n, n+1) = 2*A000108(n-1), n >= 1.
From G. C. Greubel, Oct 13 2022: (Start)
T(n, n-1) = A000027(n-2) + 2*[n<3], n >= 1.
T(n, n-2) = A000096(n-4) + 2*[n<5], n >= 2.
T(n, n-3) = A005586(n-6) + 4*[n<7] - 2*[n=3], n >= 3.
T(2*n, n) = 2*A000108(n-1) + 3*[n=0].
T(2*n-1, n-1) = T(2*n+1, n+1) = A000180(n).
T(3*n, n) = A025174(n) + [n=0]
Sum_{k=0..n} T(n, k) = 2*A063886(n-2) + [n=0] - 2*[n=1]
Sum_{k=0..n} (-1)^k * T(n, k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n, k) = A047079(n). (End)

A047079 a(n) = Sum_{i=0..floor(n/2)} A047072(i, n-2*i).

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 7, 9, 14, 23, 33, 52, 85, 127, 202, 329, 503, 804, 1307, 2027, 3250, 5277, 8263, 13276, 21539, 33957, 54638, 88595, 140373, 226108, 366481, 582865, 939622, 1522487, 2428517, 3917412, 6345929, 10145769, 16374126
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A047072, A047073, A047074.

Programs

  • Magma
    b:= func< n | n eq 0 select 1 else 2*Catalan(n-1) >;
function A(n, k)
  if k eq n then return b(n);
  elif k gt n then return Binomial(n+k-1, n) - Binomial(n+k-1, n-1);
  else return Binomial(n+k-1, k) - Binomial(n+k-1, k-1);
  end if; return A;
end function;
[(&+[A(j, n-2*j): j in [0..Floor(n/2)]]): n in [0..50]]; // G. C. Greubel, Oct 29 2022
  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==n, 2*CatalanNumber[n-1] +2*Boole[n==0], If[k>n, Binomial[n+k-1,n] -Binomial[n+k-1,n-1], Binomial[n+k-1,k] -Binomial[n+k-1, k- 1]]];
A047079[n_]:= Sum[T[j, n-2*j], {j,0,Floor[n/2]}] +Boole[n==0];
Table[A047079[n], {n,0,50}] (* G. C. Greubel, Oct 29 2022 *)
  • SageMath
    def A047072(n, k): # array
    if (k==n): return 2*catalan_number(n-1) + 2*int(n==0)
    elif (k>n): return binomial(n+k-1, n) - binomial(n+k-1, n-1)
    else: return binomial(n+k-1, k) - binomial(n+k-1, k-1)
def A047079(n): return sum( A047072(j, n-2*j) for j in range(((n+1)//2)+1) )
[A047079(n) for n in range(51)] # G. C. Greubel, Oct 29 2022

Extensions

Name improved by Sean A. Irvine, May 11 2021

A137277 Triangle of the coefficients [x^k] P_n(x) of the polynomials P_n(x) = 1/n * sum(j=0..floor(n/2), (-1)^j * binomial(n,j) * (n-4*j) * x^(n-2*j) ).

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 1, 0, 1, -6, 0, 0, 0, 1, 0, -6, 0, -1, 0, 1, 20, 0, -5, 0, -2, 0, 1, 0, 25, 0, -3, 0, -3, 0, 1, -70, 0, 28, 0, 0, 0, -4, 0, 1, 0, -98, 0, 28, 0, 4, 0, -5, 0, 1, 252, 0, -126, 0, 24, 0, 9, 0, -6, 0, 1, 0, 378, 0, -150, 0, 15, 0, 15, 0, -7, 0, 1, -924, 0, 528, 0, -165, 0, 0, 0, 22, 0, -8, 0, 1, 0, -1452
Offset: 0

Views

Author

Roger L. Bagula, Mar 13 2008

Keywords

Comments

The first four P_n(x) are the same as in A137276.
Row sums are 1, 1, 3, 2, -5, -6, 14, 20, -45, -70, 154, a signed variant of A047074.

Examples

			{1}, = 1
{0, 1}, = x
{2, 0, 1}, = 2+x^2
{0, 1, 0, 1}, = x+x^3
{-6, 0, 0, 0, 1}, = -6+x^4
{0, -6, 0, -1, 0, 1},
{20, 0, -5, 0, -2, 0, 1},
{0, 25, 0, -3,0, -3, 0, 1},
{-70, 0, 28, 0, 0, 0, -4, 0, 1},
{0, -98, 0, 28, 0,4, 0, -5, 0, 1},
{252, 0, -126, 0, 24, 0, 9, 0, -6, 0, 1}

Crossrefs

Cf. A138034.

Programs

  • Maple
    A137277 := proc(n,k) if n = 0 then 1; else add( (-1)^j*binomial(n,j)*(n-4*j)*x^(n-2*j),j=0..n/2)/n ; coeftayl(%,x=0,k) ; fi; end:
seq( seq(A137277(n,k),k=0..n),n=0..15) ;
  • Mathematica
    B[x_, n_] = If[n > 0, Sum[(-1)^p*Binomial[n,p]*(n - 4*p)*x^(n - 2*p)/ n, {p, 0, Floor[n/2]}], 1]; a = Table[CoefficientList[B[x, n], x], {n, 0, 10}]; Flatten[a]

Formula

P(0,n)=1. P_n(x) = 1/n*sum(j=0..floor(n/2), (-1)^j*binomial(n,j)*(n-4*j)*x^(n-2*j)).

Extensions

Edited by the Associate Editors of the OEIS, Aug 27 2009
Showing 1-4 of 4 results.

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