A026749 -id:A026749 - cp's OEIS frontend

A026754 a(n) = Sum{k=0..n} T(n,k), T given by A026747.

1, 2, 5, 10, 24, 48, 114, 228, 540, 1080, 2558, 5116, 12133, 24266, 57658, 115316, 274600, 549200, 1310817, 2621634, 6271788, 12543576, 30076629, 60153258, 144550655, 289101310, 696176322, 1392352644, 3359516328

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A026747, A026748, A026749, A026750, A026751, A026752, A026753, A026755, A026756, A026757.

A026747 := proc(n,k) option remember;
   if k=0 or k = n then 1;
   elif type(n,'even') and k <= n/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
   else
       procname(n-1,k-1)+procname(n-1,k) ;
   end if ;
end proc:
seq(add(A026747(n,k), k=0..n), n=0..30); # G. C. Greubel, Oct 29 2019

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[EvenQ[n] && k<=n/2, T[n-1, k -1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]; Table[Sum[T[n, k],{k,0,n}], {n,0,30}] (* G. C. Greubel, Oct 29 2019 *)

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k<=n/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[sum(T(n, k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Oct 29 2019

A026748 a(n) = T(2n,n), T given by A026747.

Original entry on oeis.org

1, 3, 11, 44, 184, 790, 3452, 15278, 68290, 307696, 1395696, 6367199, 29193025, 134442102, 621609060, 2884432810, 13428450520, 62703991531, 293606387095, 1378309455352, 6485734373020, 30586630485443, 144544075759391, 684395988590939

Offset: 0

Clark Kimberling

nonn

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

Cf. A026747, A026749, A026750, A026751, A026752, A026753, A026754, A026755, A026756, A026757.

A026747 := proc(n,k) option remember;
   if k=0 or k = n then 1;
   elif type(n,'even') and k <= n/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
   else
       procname(n-1,k-1)+procname(n-1,k) ;
   end if ;
end proc:
seq(A026747(2*n,n), n=0..30); # G. C. Greubel, Oct 29 2019

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[EvenQ[n] && k<=n/2, T[n-1, k -1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]; Table[T[2n, n], {n,0,30}] (* G. C. Greubel, Oct 29 2019 *)

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k<=n/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[T(2*n, n) for n in (0..30)] # G. C. Greubel, Oct 29 2019

A026750 a(n) = T(2n,n-2), T given by A026747.

Original entry on oeis.org

1, 9, 58, 329, 1753, 9020, 45442, 225860, 1112543, 5446607, 26550968, 129042976, 625860205, 3031021096, 14664729519, 70906318405, 342717456708, 1656208470644, 8003645557573, 38681730323747, 186985728069661, 904119336235884

Offset: 2

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 2..500

Cf. A026747, A026748, A026749, A026751, A026752, A026753, A026754, A026755, A026756, A026757.

A026747 := proc(n,k) option remember;
   if k=0 or k = n then 1;
   elif type(n,'even') and k <= n/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
   else
       procname(n-1,k-1)+procname(n-1,k) ;
   end if ;
end proc:
seq(A026747(2*n,n-2), n=2..30); # G. C. Greubel, Oct 29 2019

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[EvenQ[n] && k<=n/2, T[n-1, k -1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]; Table[T[2n, n-2], {n,2,30}] (* G. C. Greubel, Oct 29 2019 *)

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k<=n/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[T(2*n, n-2) for n in (2..30)] # G. C. Greubel, Oct 29 2019

A026751 a(n) = T(2n-1,n-1), T given by A026747.

Original entry on oeis.org

1, 4, 17, 74, 327, 1461, 6584, 29879, 136391, 625731, 2883357, 13338421, 61920497, 288368511, 1346873365, 6307694990, 29613690966, 139352892908, 657163401162, 3105304341356, 14701236957028, 69722518168060, 331220099616432

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..500

Cf. A026747, A026748, A026749, A026750, A026752, A026753, A026754, A026755, A026756, A026757.

A026747 := proc(n,k) option remember;
   if k=0 or k = n then 1;
   elif type(n,'even') and k <= n/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
   else
       procname(n-1,k-1)+procname(n-1,k) ;
   end if ;
end proc:
seq(A026747(2*n-1,n-1), n=1..30); # G. C. Greubel, Oct 29 2019

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[EvenQ[n] && k<=n/2, T[n-1, k -1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]; Table[T[2n-1, n-1], {n,30}] (* G. C. Greubel, Oct 29 2019 *)

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k<=n/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[T(2*n-1, n-1) for n in (1..30)] # G. C. Greubel, Oct 29 2019

A026752 a(n) = T(2n-1,n-2), T given by A026747.

Original entry on oeis.org

1, 7, 39, 201, 1000, 4885, 23621, 113543, 543895, 2600204, 12417829, 59278440, 282969385, 1351124510, 6454283276, 30849969965, 147555219782, 706274470775, 3383203356648, 16219148141581, 77817618006364, 373661751926702

Offset: 2

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 2..500

Cf. A026747, A026748, A026749, A026750, A026751, A026753, A026754, A026755, A026756, A026757.

A026747 := proc(n,k) option remember;
   if k=0 or k = n then 1;
   elif type(n,'even') and k <= n/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
   else
       procname(n-1,k-1)+procname(n-1,k) ;
   end if ;
end proc:
seq(A026747(2*n-1,n-2), n=2..30); # G. C. Greubel, Oct 29 2019

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[EvenQ[n] && k<=n/2, T[n-1, k -1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]; Table[T[2n-1, n-2], {n,2,30}] (* G. C. Greubel, Oct 29 2019 *)

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k<=n/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[T(2*n-1, n-2) for n in (2..30)] # G. C. Greubel, Oct 29 2019

A026753 a(n) = T(n, floor(n/2)), T given by A026747.

Original entry on oeis.org

1, 1, 3, 4, 11, 17, 44, 74, 184, 327, 790, 1461, 3452, 6584, 15278, 29879, 68290, 136391, 307696, 625731, 1395696, 2883357, 6367199, 13338421, 29193025, 61920497, 134442102, 288368511, 621609060, 1346873365, 2884432810

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A026747, A026748, A026749, A026750, A026751, A026752, A026754, A026755, A026756, A026757.

A026747 := proc(n,k) option remember;
   if k=0 or k = n then 1;
   elif type(n,'even') and k <= n/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
   else
       procname(n-1,k-1)+procname(n-1,k) ;
   end if ;
end proc:
seq(A026747(n,floor(n/2)), n=0..30); # G. C. Greubel, Oct 29 2019

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[EvenQ[n] && k<=n/2, T[n-1, k -1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]; Table[T[n, Floor[n/2]], {n,0,30}] (* G. C. Greubel, Oct 29 2019 *)

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k<=n/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[T(n, floor(n/2)) for n in (0..30)] # G. C. Greubel, Oct 29 2019

A026755 a(n) = Sum_{k=0..floor(n/2)} T(n,k), T given by A026747.

Original entry on oeis.org

1, 1, 4, 5, 18, 25, 84, 124, 398, 612, 1901, 3012, 9126, 14800, 43968, 72658, 212417, 356544, 1028520, 1749344, 4989477, 8583258, 24244139, 42121079, 117973702, 206754379, 574811040, 1015179978, 2803969443, 4986329826

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A026747, A026748, A026749, A026750, A026751, A026752, A026753, A026754, A026756, A026757.

A026747 := proc(n,k) option remember;
   if k=0 or k = n then 1;
   elif type(n,'even') and k <= n/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
   else
       procname(n-1,k-1)+procname(n-1,k) ;
   end if ;
end proc:
seq(add(A026747(n,k), k=0..floor(n/2)), n=0..30); # G. C. Greubel, Oct 29 2019

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[EvenQ[n] && k<=n/2, T[n-1, k -1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]; Table[Sum[T[n, k], Floor[n/2]], {n,0,30}] (* G. C. Greubel, Oct 29 2019 *)

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k<=n/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[sum(T(n, k) for k in (0..floor(n/2))) for n in (0..30)] # G. C. Greubel, Oct 29 2019

A026756 a(n) = Sum_{i=0..n} Sum_{j=0..n} T(i,j), T given by A026747.

Original entry on oeis.org

1, 3, 8, 18, 42, 90, 204, 432, 972, 2052, 4610, 9726, 21859, 46125, 103783, 219099, 493699, 1042899, 2353716, 4975350, 11247138, 23790714, 53867343, 114020601, 258571256, 547672566, 1243848888, 2636201532, 5995717860

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A026747, A026748, A026749, A026750, A026751, A026752, A026753, A026754, A026755, A026757.

A026747 := proc(n,k) option remember;
   if n<0 then 0;
   elif k=0 or k = n then 1;
   elif type(n,'even') and k <= n/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
   else
       procname(n-1,k-1)+procname(n-1,k) ;
   end if ;
end proc:
seq(add(add(A026747(i,j), j=0..n), i=0..n), n=0..30); # G. C. Greubel, Oct 29 2019

T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[EvenQ[n] && k<=n/2, T[n-1, k -1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]]; Table[Sum[T[i, j], {i,0,n},{j,0,n}], {n,0,30}] (* G. C. Greubel, Oct 29 2019 *)

@CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k<=n/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[sum(sum(T(i,j) for j in (0..n)) for i in (0..n)) for n in (0..30)] # G. C. Greubel, Oct 29 2019

A026757 a(n) = Sum_{k=0..floor(n/2)} T(n-k,k), T given by A026747.

Original entry on oeis.org

1, 1, 2, 4, 6, 11, 20, 32, 58, 102, 169, 302, 527, 888, 1573, 2741, 4661, 8215, 14316, 24481, 43023, 74998, 128747, 225867, 393838, 678047, 1188201, 2072239, 3575728, 6261248, 10921278, 18879372, 33040083, 57637061

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A026747, A026748, A026749, A026750, A026751, A026752, A026753, A026754, A026755, A026756.

A026747 := proc(n,k) option remember;
   if k=0 or k = n then 1;
   elif type(n,'even') and k <= n/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
   else
       procname(n-1,k-1)+procname(n-1,k) ;
   end if ;
end proc:
seq(add(A026747(n-k,k), k=0..floor(n/2)), n=0..30); # G. C. Greubel, Oct 29 2019

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[EvenQ[n] && k<=n/2, T[n-1, k -1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]; Table[Sum[T[n-k, k], {k,0,Floor[n/2]}], {n,0,30}] (* G. C. Greubel, Oct 29 2019 *)

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k<=n/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[sum(T(n-k, k) for k in (0..floor(n/2))) for n in (0..30)] # G. C. Greubel, Oct 29 2019

A210381 Triangle by rows, derived from the beheaded Pascal's triangle, A074909.

Original entry on oeis.org

1, 0, 2, 0, 1, 3, 0, 1, 3, 4, 0, 1, 4, 6, 5, 0, 1, 5, 10, 10, 6, 0, 1, 6, 15, 20, 15, 7, 0, 1, 7, 21, 35, 35, 21, 8, 0, 1, 8, 28, 56, 70, 56, 28, 9, 0, 1, 9, 36, 84, 126, 126, 84, 36, 10, 0, 1, 10, 45, 120, 210, 252, 210, 120, 45, 11

Offset: 0

Roger L. Bagula and Gary W. Adamson, Mar 20 2012

Row sums of the triangle = 2^n.
Let the triangle = an infinite lower triangular matrix, M.  Then M * The Bernoulli numbers, A027641/A027642  as a vector V = [1, -1, 0, 0, 0,...].  M * the Bernoulli sequence variant starting [1, 1/2, 1/6,...] = [1, 1, 1,...].  M * 2^n: [1, 2, 4, 8,...] = A027649.  M * 3^n = A255463; while M * [1, 2, 3,...] = A047859, and M * A027649 = A027650.
Row sums of powers of the triangle generate the Poly-Bernoulli number sequences shown in the array of A099594. - Gary W. Adamson, Mar 21 2012
Triangle T(n,k) given by (0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 25 2012

			{1},
{0, 2},
{0, 1, 3},
{0, 1, 3, 4},
{0, 1, 4, 6, 5},
{0, 1, 5, 10, 10, 6},
{0, 1, 6, 15, 20, 15, 7},
{0, 1, 7, 21, 35, 35, 21, 8},
{0, 1, 8, 28, 56, 70, 56, 28, 9},
{0, 1, 9, 36, 84, 126, 126, 84, 36, 10},
{0, 1, 10, 45, 120, 210, 252, 210, 120, 45, 11}
...

Konrad Knopp, Elements of the Theory of Functions, Dover, 1952,pp 117-118.

Cf. A074909, A255463, A026749, A047859, A027650

Cf. A099594.

t2[n_, m_] = If[m - 1 <= n, Binomial[n, m - 1], 0];
O2 = Table[Table[If[n == m, t2[n, m] + 1, t2[n, m]], {m, 0, n}], {n, 0, 10}];
Flatten[O2]

Partial differences of the beheaded Pascal's triangle A074909 starting from the top, by columns.
G.f.: (1-x)/(1-x-2*y*x+y*x^2+y^2*x^2). - Philippe Deléham, Mar 25 2012
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(2,1) = 1, T(1,0) = T(2,0) = 0, T(1,1) = 2, T(2,2) = 3 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 25 2012

cp's OEIS Frontend

A026754 a(n) = Sum{k=0..n} T(n,k), T given by A026747.

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A026748 a(n) = T(2n,n), T given by A026747.

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A026750 a(n) = T(2n,n-2), T given by A026747.

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Maple

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Sage

A026751 a(n) = T(2n-1,n-1), T given by A026747.

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Maple

Mathematica

Sage

A026752 a(n) = T(2n-1,n-2), T given by A026747.

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Author

Keywords

Links

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Programs

Maple

Mathematica

Sage

A026753 a(n) = T(n, floor(n/2)), T given by A026747.

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Programs

Maple

Mathematica

Sage

A026755 a(n) = Sum_{k=0..floor(n/2)} T(n,k), T given by A026747.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Sage

A026756 a(n) = Sum_{i=0..n} Sum_{j=0..n} T(i,j), T given by A026747.

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Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica