A118346 -id:A118346 - cp's OEIS frontend

A118349 Convolution triangle, read by rows, where diagonals are successive self-convolutions of A118346.

1, 1, 0, 1, 1, 0, 1, 2, 5, 0, 1, 3, 11, 30, 0, 1, 4, 18, 70, 201, 0, 1, 5, 26, 121, 487, 1445, 0, 1, 6, 35, 184, 873, 3592, 10900, 0, 1, 7, 45, 260, 1375, 6606, 27600, 85128, 0, 1, 8, 56, 350, 2010, 10672, 51728, 218566, 682505, 0, 1, 9, 68, 455, 2796, 15996, 85182, 415629, 1771367, 5585115, 0

Offset: 0

Paul D. Hanna, Apr 26 2006

A118346 equals the central terms of pendular triangle A118345 and the diagonals of this triangle form the semi-diagonals of the triangle A118345.

			Show: T(n,k) = T(n-1,k) - 2*T(n-1,k-1) + 2*T(n,k-1) + T(n+1,k-1)
at n=8,k=4: T(8,4) = T(7,4) - 2*T(7,3) + 2*T(8,3) + T(9,3)
or: 1375 = 873 - 2*184 + 2*260 + 350.
Triangle begins:
  1;
  1, 0;
  1, 1,  0;
  1, 2,  5,   0;
  1, 3, 11,  30,    0;
  1, 4, 18,  70,  201,     0;
  1, 5, 26, 121,  487,  1445,     0;
  1, 6, 35, 184,  873,  3592, 10900,      0;
  1, 7, 45, 260, 1375,  6606, 27600,  85128,       0;
  1, 8, 56, 350, 2010, 10672, 51728, 218566,  682505,       0;
  1, 9, 68, 455, 2796, 15996, 85182, 415629, 1771367, 5585115, 0;

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

Cf. A118346, A118345.

T:= proc(n, k) option remember;
      if k<0 or  k>n then 0;
    elif k=0 then 1;
    elif k=n then 0;
    else T(n-1, k) -2*T(n-1, k-1) +2*T(n, k-1) +T(n+1, k-1);
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 17 2021

T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, 1, If[k==n, 0, T[n-1, k] -2*T[n-1, k-1] +2*T[n, k-1] +T[n+1, k-1] ]]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 17 2021 *)

{T(n,k)=polcoeff((serreverse(x*(1-2*x+sqrt((1-2*x)*(1-6*x)+x*O(x^k)))/2/(1-2*x))/x)^(n-k),k)}

@CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (k==0): return 1
    elif (k==n): return 0
    else: return T(n-1, k) -2*T(n-1, k-1) +2*T(n, k-1) +T(n+1, k-1)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 17 2021

Since g.f. G=G(x) of A118346 satisfies: G = 1 - 2*x*G + 2*x*G^2 + x*G^3 then T(n,k) = T(n-1,k) - 2*T(n-1,k-1) + 2*T(n,k-1) + T(n+1,k-1). Also, a recurrence involving antidiagonals is: T(n,k) = T(n-1,k) + Sum_{j=1..k} [3*T(n-1+j,k-j) - 2*T(n-2+j,k-j)] for n>k>=0.

A118347 Semi-diagonal (one row below central terms) of pendular triangle A118345 and equal to the self-convolution of the central terms (A118346).

Original entry on oeis.org

1, 2, 11, 70, 487, 3592, 27600, 218566, 1771367, 14621410, 122495659, 1038934480, 8903129300, 76970244560, 670507216168, 5879770542870, 51861650744071, 459804626981158, 4095433894576785, 36628711884398086, 328824295880947471

Offset: 0

Paul D. Hanna, Apr 26 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A118345, A118346, A118348, A118349.

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (Reversion( x*(1-2*x +Sqrt((1-2*x)*(1-6*x)))/(2*(1-2*x)) )/x)^2 )); // G. C. Greubel, Mar 17 2021

CoefficientList[(InverseSeries[Series[x*(1-2*x +Sqrt[(1-2*x)*(1-6*x)])/(2*(1-2*x)), {x, 0, 30}]]/x)^2, x] (* G. C. Greubel, Mar 17 2021 *)

{a(n) = polcoeff( (serreverse(x*(1-2*x+sqrt((1-2*x)*(1-6*x)+x*O(x^n)))/(2*(1-2*x)))/x)^2,n)}

def A118347_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (( x*(1-2*x +sqrt((1-2*x)*(1-6*x)))/(2*(1-2*x)) ).reverse()/x)^2 ).list()
A118347_list(31) # G. C. Greubel, Mar 17 2021

G.f.: ( series_inverse( x*(1-2*x +sqrt((1-2*x)*(1-6*x)))/(2*(1-2*x)) )/x )^2.

A118348 Semi-diagonal (two rows below central terms) of pendular triangle A118345 and equal to the self-convolution cube of the central terms (A118346).

Original entry on oeis.org

1, 3, 18, 121, 873, 6606, 51728, 415629, 3407391, 28388847, 239675406, 2045980440, 17629939980, 153142537440, 1339599358944, 11789960853293, 104327344928619, 927627432162129, 8283625668834238, 74259685465582569, 668054892245119353

Offset: 0

Paul D. Hanna, Apr 26 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A118345, A118346, A118347, A118349.

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (Reversion( x*(1-2*x +Sqrt((1-2*x)*(1-6*x)))/(2*(1-2*x)) )/x)^3 )); // G. C. Greubel, Mar 17 2021

CoefficientList[(InverseSeries[Series[x*(1-2*x +Sqrt[(1-2*x)*(1-6*x)])/(2*(1-2*x)), {x, 0, 30}]]/x)^3, x] (* G. C. Greubel, Mar 17 2021 *)

{a(n) = polcoeff( (serreverse(x*(1-2*x+sqrt((1-2*x)*(1-6*x)+x*O(x^n)))/(2*(1-2*x)))/x)^3, n)}

def A118347_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (( x*(1-2*x +sqrt((1-2*x)*(1-6*x)))/(2*(1-2*x)) ).reverse()/x)^3 ).list()
A118347_list(31) # G. C. Greubel, Mar 17 2021

G.f.: ( series_inverse( x*(1-2*x +sqrt((1-2*x)*(1-6*x)))/(2*(1-2*x)) )/x )^3.

A118345 Pendular triangle, read by rows, where row n is formed from row n-1 by the recurrence: if n > 2k, T(n,k) = T(n,n-k) + T(n-1,k), else T(n,k) = T(n,n-1-k) + 2*T(n-1,k), for n>=k>=0, with T(n,0) = 1 and T(n,n) = 0^n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 5, 1, 0, 1, 4, 11, 6, 1, 0, 1, 5, 18, 30, 7, 1, 0, 1, 6, 26, 70, 40, 8, 1, 0, 1, 7, 35, 121, 201, 51, 9, 1, 0, 1, 8, 45, 184, 487, 286, 63, 10, 1, 0, 1, 9, 56, 260, 873, 1445, 386, 76, 11, 1, 0, 1, 10, 68, 350, 1375, 3592, 2147, 502, 90, 12, 1, 0

Offset: 0

Paul D. Hanna, Apr 26 2006

See A118340 for definition of pendular triangles and pendular sums.

			Row 6 equals the pendular sums of row 5:
  [1,  4, 11,  6,  1,  0], where the pendular sums proceed as follows:
  [1, __, __, __, __, __]: T(6,0) = T(5,0) = 1;
  [1, __, __, __, __,  1]: T(6,5) = T(6,0) + 2*T(5,5) = 1 + 2*0 = 1;
  [1,  5, __, __, __,  1]: T(6,1) = T(6,5) + T(5,1) = 1 + 4 = 5;
  [1,  5, __, __,  7,  1]: T(6,4) = T(6,1) + 2*T(5,4) = 5 + 2*1 = 7;
  [1,  5, 18, __,  7,  1]: T(6,2) = T(6,4) + T(5,2) = 7 + 11 = 18;
  [1,  5, 18, 30,  7,  1]: T(6,3) = T(6,2) + 2*T(5,3) = 18 + 2*6 = 30;
  [1,  5, 18, 30,  7,  1, 0] finally, append a zero to obtain row 6.
Triangle begins:
  1;
  1,  0;
  1,  1,  0;
  1,  2,  1,   0;
  1,  3,  5,   1,    0;
  1,  4, 11,   6,    1,    0;
  1,  5, 18,  30,    7,    1,    0;
  1,  6, 26,  70,   40,    8,    1,   0;
  1,  7, 35, 121,  201,   51,    9,   1,  0;
  1,  8, 45, 184,  487,  286,   63,  10,  1,  0;
  1,  9, 56, 260,  873, 1445,  386,  76, 11,  1, 0;
  1, 10, 68, 350, 1375, 3592, 2147, 502, 90, 12, 1, 0; ...
Central terms are T(2*n,n) = A118346(n);
semi-diagonals form successive self-convolutions of the central terms:
T(2*n+1,n) = A118347(n) = [A118346^2](n),
T(2*n+2,n) = A118348(n) = [A118346^3](n).

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A118346, A118347, A118348, A118349, A118340.

Cf. A167763 (p=0), A118340 (p=1), this sequence (p=2), A118350 (p=3).

function T(n,k,p)
  if k lt 0 or n lt k then return 0;
  elif k eq 0 then return 1;
  elif k eq n then return 0;
  elif n gt 2*k then return T(n,n-k,p) + T(n-1,k,p);
  else return T(n,n-k-1,p) + p*T(n-1,k,p);
  end if;
  return T;
end function;
[T(n,k,2): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 17 2021

T[n_, k_, p_]:= T[n,k,p] = If[nG. C. Greubel, Feb 17 2021 *)

T(n,k)=if(n2*k,T(n,n-k)+T(n-1,k),T(n,n-1-k)+2*T(n-1,k)))))

@CachedFunction
def T(n, k, p):
    if (k<0 or n2*k): return T(n,n-k,p) + T(n-1,k,p)
    else: return T(n, n-k-1, p) + p*T(n-1, k, p)
flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 17 2021

T(2*n+m,n) = [A118346^(m+1)](n), i.e., the m-th lower semi-diagonal forms the self-convolution (m+1)-power of A118346.

A336707 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (1/n) * Sum_{j=1..n} 2^(n-j) * binomial(n,j) * binomial(n+(k-1)*j,j-1) for n > 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 11, 20, 1, 1, 5, 19, 45, 72, 1, 1, 6, 30, 100, 197, 272, 1, 1, 7, 44, 201, 562, 903, 1064, 1, 1, 8, 61, 364, 1445, 3304, 4279, 4272, 1, 1, 9, 81, 605, 3249, 10900, 20071, 20793, 17504, 1, 1, 10, 104, 940, 6502, 30526, 85128, 124996, 103049, 72896

Offset: 0

Seiichi Manyama, Aug 01 2020

			Square array begins:
    1,   1,    1,     1,     1,     1,      1, ...
    1,   1,    1,     1,     1,     1,      1, ...
    2,   3,    4,     5,     6,     7,      8, ...
    6,  11,   19,    30,    44,    61,     81, ...
   20,  45,  100,   201,   364,   605,    940, ...
   72, 197,  562,  1445,  3249,  6502,  11857, ...
  272, 903, 3304, 10900, 30526, 73723, 158034, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-3 give: A071356(n-1), A001003, A007564, A118346.
Main diagonal gives A336712.
Cf. A336575, A336706, A336708, A336709.

T[0, k_] := 1; T[n_, k_] := Sum[2^(n - j) * Binomial[n, j] * Binomial[n + (k - 1)*j, j - 1], {j, 1, n}] / n; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 01 2020 *)

{T(n, k) = if(n==0, 1, sum(j=1, n, 2^(n-j)*binomial(n, j)*binomial(n+(k-1)*j, j-1))/n)}

{T(n, k) = local(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^k/(1-2*x*A)); polcoef(A, n)}

G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x)^k / (1 - 2 * x * A_k(x)).

cp's OEIS Frontend

A118349 Convolution triangle, read by rows, where diagonals are successive self-convolutions of A118346.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Formula

A118347 Semi-diagonal (one row below central terms) of pendular triangle A118345 and equal to the self-convolution of the central terms (A118346).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A118348 Semi-diagonal (two rows below central terms) of pendular triangle A118345 and equal to the self-convolution cube of the central terms (A118346).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A118345 Pendular triangle, read by rows, where row n is formed from row n-1 by the recurrence: if n > 2k, T(n,k) = T(n,n-k) + T(n-1,k), else T(n,k) = T(n,n-1-k) + 2*T(n-1,k), for n>=k>=0, with T(n,0) = 1 and T(n,n) = 0^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A336707 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (1/n) * Sum_{j=1..n} 2^(n-j) * binomial(n,j) * binomial(n+(k-1)*j,j-1) for n > 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula