A055817 -id:A055817 - cp's OEIS frontend

A055807 Triangle T read by rows: T(i,j) = R(i-j,j), where R(i,0) = 1 for i >= 0, R(0,j) = 0 for j >= 1, and R(i,j) = Sum_{h=0..i-1, k=0..j} R(h,k) for i >= 1 and j >= 1.

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 7, 4, 1, 0, 1, 15, 12, 5, 1, 0, 1, 31, 32, 18, 6, 1, 0, 1, 63, 80, 56, 25, 7, 1, 0, 1, 127, 192, 160, 88, 33, 8, 1, 0, 1, 255, 448, 432, 280, 129, 42, 9, 1, 0, 1, 511, 1024, 1120, 832, 450, 180, 52, 10, 1, 0, 1, 1023

Offset: 0

Clark Kimberling, May 28 2000

Formatted as a triangular array, it is [1, 0, 1, 1, 0, 0, 0, 0, 0, ...] DELTA [0, 1, 0, -1, 1, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 05 2006

The square array (R(n,k): n,k >= 0) referred to in the name of the sequence is actually A050143. - Petros Hadjicostas, Feb 13 2021

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,   0;
  1,   1,   0;
  1,   3,   1,   0;
  1,   7,   4,   1,   0;
  1,  15,  12,   5,   1,   0;
  1,  31,  32,  18,   6,   1,  0;
  1,  63,  80,  56,  25,   7,  1, 0;
  1, 127, 192, 160,  88,  33,  8, 1, 0;
  1, 255, 448, 432, 280, 129, 42, 9, 1, 0;
  ...
Florez et al. (2019) give the triangle in this form:
    1,    0,    0,   0,   0,   0,  0,  0, 0, ...
    3,    1,    0,   0,   0,   0,  0,  0, 0, ...
    7,    4,    1,   0,   0,   0,  0,  0, 0, ...
   15,   12,    5,   1,   0,   0,  0,  0, 0, ...
   31,   32,   18,   6,   1,   0,  0,  0, 0, ...
   63,   80,   56,  25,   7,   1,  0,  0, 0, ...
  127,  192,  160,  88,  33,   8,  1,  0, 0, ...
  255,  448,  432, 280, 129,  42,  9,  1, 0, ...
  511, 1024, 1120, 832, 450, 180, 52, 10, 1, ...
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Enumerating several aspects of non-decreasing Dyck paths, Discrete Mathematics Vol. 342, Issue 11 (2019), 3079-3097. See page 3091. Gives the triangle in a slightly different form (see the Examples section).
Clark Kimberling, Path-counting and Fibonacci numbers, Fib. Quart. 40(4) (2002) 328-338, Example 3A.

Rows sums: A001519 (odd-indexed Fibonacci numbers).
Cf. A050143, A050147, A050148, A111516.
Cf. A055809, A055810, A055811, A055815, A055816, A055817.

T:= function(i,j)
    if j=0 then return 1;
    elif i=0 then return 0;
    else return Sum([0..i-1], h-> Sum([0..j], m-> T(h,m) ));
    fi; end;
Flat(List([0..12], n-> List([0..n], k-> T(n-k,k) ))); # G. C. Greubel, Jan 23 2020

function T(i,j)
  if j eq 0 then return 1;
  elif i eq 0 then return 0;
  else return (&+[(&+[T(h,m): m in [0..j]]): h in [0..i-1]]);
  end if; return T; end function;
[T(n-k,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 23 2020

T:= proc(i, j) option remember;
      if j=0 then 1
    elif i=0 then 0
    else add(add(T(h,m), m=0..j), h=0..i-1)
      fi; end:
seq(seq(T(n-k, k), k=0..n), n=0..12); # G. C. Greubel, Jan 23 2020

T[i_, j_]:= T[i, j]= If[j==0, 1, If[i==0, 0, Sum[T[h, m], {h,0,i-1}, {m,0,j}]]]; Table[T[n-k, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 23 2020 *)

T(i,j) = if(j==0, 1, if(i==0, 0, sum(h=0,i-1, sum(m=0,j, T(h,m) ))));
for(n=0,12, for(k=0, n, print1(T(n-k,k), ", "))) \\ G. C. Greubel, Jan 23 2020

@CachedFunction
def T(i, j):
    if j==0: return 1
    elif i==0: return 0
    else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
[[T(n-k, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 23 2020

T(2*n,n) = A050146(n).
G.f.: (1-2*x)*(1-x*y)/((1-x)*(1-x*y-2*x+x^2*y)). - R. J. Mathar, Aug 11 2015
From Petros Hadjicostas, Feb 13 2021: (Start)
T(n,k) = A050143(n-k, k) for 0 <= k <= n.
T(n,k) = (n-k)*hypergeom([-n + k + 1, k], [2], -1) = Sum_{s=1..n-k} binomial(n-k,s)*binomial(s+k-2,k-1) for 1 <= k <= n.
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) for 2 <= k <= n-1 with initial conditions T(n,0) = 1 for n >= 0, T(n,n) = 0 for n >= 1, and T(n,1) = 2^(n-1) - 1 for n >= 2. (End)

A055809 a(n) = T(n,n-4), array T as in A055807.

Original entry on oeis.org

1, 15, 32, 56, 88, 129, 180, 242, 316, 403, 504, 620, 752, 901, 1068, 1254, 1460, 1687, 1936, 2208, 2504, 2825, 3172, 3546, 3948, 4379, 4840, 5332, 5856, 6413, 7004, 7630, 8292, 8991, 9728, 10504, 11320, 12177, 13076

Offset: 4

Clark Kimberling, May 28 2000

If Y_i (i=1,2,3,4) are 2-blocks of an n-set X then, for n>=8, a(n-2) is the number of (n-3)-subsets of X intersecting each Y_i (i=1,2,3,4). - Milan Janjic, Nov 09 2007

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

Cf. A055807, A055810, A055811, A055815, A055816, A055817.

Concatenation([1], List([5..50], n-> n*(n^2 +3*n -22)/6 )); # G. C. Greubel, Jan 23 2020

[1] cat [n*(n^2 +3*n -22)/6: n in [5..50]]; // G. C. Greubel, Jan 23 2020

seq( `if`(n=4, 1, n*(n^2 +3*n -22)/6), n=4..50); # G. C. Greubel, Jan 23 2020

f[n_]:=Sum[i+i^2-8,{i,1,n}]/2;Table[f[n],{n,5,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 08 2010 *)
Table[If[n==4, 1, n*(n^2 +3*n -22)/6], {n,4,50}] (* G. C. Greubel, Jan 23 2020 *)

Vec(x^4*(1 + 11*x - 22*x^2 + 14*x^3 - 3*x^4)/(1-x)^4 + O(x^50)) \\ Michel Marcus, Jan 10 2015

vector(50, n, my(m=n+3); if(m==4, 1, m*(m^2 +3*m -22)/6)) \\ G. C. Greubel, Jan 23 2020

[1]+[n*(n^2 +3*n -22)/6 for n in (5..50)] # G. C. Greubel, Jan 23 2020

For n>4, a(n) = n*(n^2 + 3*n - 22)/6.
G.f.: x^4*(1 + 11*x - 22*x^2 + 14*x^3 - 3*x^4)/(1-x)^4. - Colin Barker, Feb 22 2012
E.g.f.: x*(72 +48*x +8*x^2 -3*x^2 + (-72 +24*x +4*x^2)*exp(x))/24. - G. C. Greubel, Jan 23 2020

A055810 a(n) = T(n,n-5), array T as in A055807.

Original entry on oeis.org

1, 31, 80, 160, 280, 450, 681, 985, 1375, 1865, 2470, 3206, 4090, 5140, 6375, 7815, 9481, 11395, 13580, 16060, 18860, 22006, 25525, 29445, 33795, 38605, 43906, 49730, 56110, 63080, 70675, 78931, 87885, 97575

Offset: 5

Clark Kimberling, May 28 2000

G. C. Greubel, Table of n, a(n) for n = 5..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A055807, A055809, A055811, A055815, A055816, A055817.

Concatenation([1], List([6..40], n-> (240 -54*n -49*n^2 +6*n^3 +n^4)/24 )); # G. C. Greubel, Jan 23 2020

[1] cat [(240 -54*n -49*n^2 +6*n^3 +n^4)/24: n in [6..40]]; // G. C. Greubel, Jan 23 2020

seq( `if`(n=5, 1, (240 -54*n -49*n^2 +6*n^3 +n^4)/24), n=5..40); # G. C. Greubel, Jan 23 2020

Table[If[n==5, 1, (240 -54*n -49*n^2 +6*n^3 +n^4)/24], {n,5,40}] (* G. C. Greubel, Jan 23 2020 *)

vector(40, n, my(m=n+4); if(m==5, 1, (240 -54*m -49*m^2 +6*m^3 +m^4)/24)) \\ G. C. Greubel, Jan 23 2020

[1]+[(240 -54*n -49*n^2 +6*n^3 +n^4)/24 for n in (6..40)] # G. C. Greubel, Jan 23 2020

G.f.: x^5*(1 +26*x -65*x^2 +60*x^3 -25*x^4 +4*x^5)/(1-x)^5. - Colin Barker, Feb 22 2012
From G. C. Greubel, Jan 23 2020: (Start)
a(n) = (240 -54*n -49*n^2 +6*n^3 +n^4)/24 for n > 5, with a(5) = 1.
E.g.f.: (-1200 -720*x +100*x^3 +25*x^4 -4*x^5 + (1200 -480*x -120*x^2 +60*x^3  +5*x^4)*exp(x))/120. (End)

A055811 a(n) = T(n,n-6), array T as in A055807.

Original entry on oeis.org

1, 63, 192, 432, 832, 1452, 2364, 3653, 5418, 7773, 10848, 14790, 19764, 25954, 33564, 42819, 53966, 67275, 83040, 101580, 123240, 148392, 177436, 210801, 248946, 292361, 341568, 397122, 459612, 529662

Offset: 6

Clark Kimberling, May 28 2000

G. C. Greubel, Table of n, a(n) for n = 6..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A055807, A055809, A055810, A055815, A055816, A055817.

Concatenation([1], List([7..30], n-> n*(1584 -310*n -85*n^2 +10*n^3 +n^4)/120 )); # G. C. Greubel, Jan 23 2020

[1] cat [n*(1584 -310*n -85*n^2 +10*n^3 +n^4)/120: n in [7.30]]; // G. C. Greubel, Jan 23 2020

seq( `if`(n=6, 1, n*(1584 -310*n -85*n^2 +10*n^3 +n^4)/120), n=6..30); # G. C. Greubel, Jan 23 2020

Table[If[n==6,1, n*(1584 -310*n -85*n^2 +10*n^3 +n^4)/120], {n,6,30}] (* G. C. Greubel, Jan 23 2020 *)

vector(25, n, my(m=n+5); if(m==6,1, m*(1584 -310*m -85*m^2 +10*m^3 +m^4)/120) ) \\ G. C. Greubel, Jan 23 2020

[1]+[n*(1584 -310*n -85*n^2 +10*n^3 +n^4)/120 for n in (7..30)] # G. C. Greubel, Jan 23 2020

From G. C. Greubel, Jan 23 2020: (Start)
a(n) = n*(1584 - 310*n - 85*n^2 + 10*n^3 + n^4)/120 for n > 6, with a(6) = 1.
G.f.: x^6*(1 + 57*x - 171*x^2 + 205*x^3 - 125*x^4 + 39*x^5 - 5*x^6)/(1-x)^6.
E.g.f.: (-1)*x*(7200 +4320*x +720*x^2 -120*x^3 -54*x^4 +5*x^5 - (7200 -2880*x + 120*x^3 + 6*x^4)*exp(x))/720. (End)

A055815 a(n) = T(2*n+3,n), array T as in A055807.

Original entry on oeis.org

1, 15, 80, 432, 2352, 12896, 71136, 394400, 2196128, 12273648, 68811184, 386838480, 2179890000, 12309739968, 69641542848, 394643939904, 2239678552640, 12727572969680, 72415319422992, 412470467298032

Offset: 0

Clark Kimberling, May 28 2000

nonn

G. C. Greubel, Table of n, a(n) for n = 0..250

Apart from the offset the same as A050149. - R. J. Mathar, Oct 13 2008

Cf. A055807, A055809, A055810, A055811, A055816, A055817.

T:= proc(i, j) option remember;
      if j=0 then 1
    elif i=0 then 0
    else add(add(T(h,m), m=0..j), h=0..i-1)
  fi; end:
seq(T(n+3, n), n=0..20); # G. C. Greubel, Jan 23 2020

T[i_, j_]:= T[i, j]= If[j==0, 1, If[i==0, 0, Sum[T[h, m], {h,0,i-1}, {m,0,j}]]]; Table[T[n+3, n], {n,0,20}] (* G. C. Greubel, Jan 23 2020 *)

@CachedFunction
def T(i, j):
    if (j==0): return 1
    elif (i==0): return 0
    else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
[T(n+3, n) for n in (0..20)] # G. C. Greubel, Jan 23 2020

a(n) = (n+3)*hypergeom([-n-2, n], [2], -1) = Sum_{s=1..n+3} binomial(n+3,s) * binomial(s+n-2,n-1) for n >= 1. - Petros Hadjicostas, Feb 13 2021

A055816 a(n) = T(2*n+4,n), array T as in A055807.

Original entry on oeis.org

1, 31, 192, 1120, 6400, 36288, 205184, 1159488, 6554880, 37088480, 210075712, 1191254688, 6762782208, 38434677120, 218663320320, 1245254943872, 7098135387648, 40495661150112, 231220652273600, 1321222104326880

Offset: 0

Clark Kimberling, May 28 2000

nonn

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A055807, A055809, A055810, A055811, A055815, A055817.

T:= proc(i, j) option remember;
      if j=0 then 1
    elif i=0 then 0
    else add(add(T(h,m), m=0..j), h=0..i-1)
  fi; end:
seq(T(n+4, n), n=0..20); # G. C. Greubel, Jan 23 2020

T[i_, j_]:= T[i, j]= If[j==0, 1, If[i==0, 0, Sum[T[h, m], {h,0,i-1}, {m,0,j}]]]; Table[T[n+4, n], {n,0,20}] (* G. C. Greubel, Jan 23 2020 *)

@CachedFunction
def T(i, j):
    if (j==0): return 1
    elif (i==0): return 0
    else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
[T(n+4, n) for n in (0..20)] # G. C. Greubel, Jan 23 2020

a(n) = (n+4)*hypergeom([-n -3, n], [2], -1) = Sum_{s=1..n+4} binomial(n+4,s)*binomial(s+n-2,n-1) for n >= 1. - Petros Hadjicostas, Feb 13 2021

cp's OEIS Frontend

A055807 Triangle T read by rows: T(i,j) = R(i-j,j), where R(i,0) = 1 for i >= 0, R(0,j) = 0 for j >= 1, and R(i,j) = Sum_{h=0..i-1, k=0..j} R(h,k) for i >= 1 and j >= 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A055809 a(n) = T(n,n-4), array T as in A055807.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

A055810 a(n) = T(n,n-5), array T as in A055807.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A055811 a(n) = T(n,n-6), array T as in A055807.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A055815 a(n) = T(2*n+3,n), array T as in A055807.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A055816 a(n) = T(2*n+4,n), array T as in A055807.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple