A026550 -id:A026550 - cp's OEIS frontend

A026536 Irregular triangular array T read by rows: T(i,0 ) = T(i,2i) = 1 for i >= 0; T(i,1) = T(i,2i-1) = floor(i/2) for i >= 1; for even n >= 2, T(i,j) = T(i-1,j-2) + T(i-1,j-1) + T(i-1,j) for j = 2..2i-2, for odd n >= 3, T(i,j) = T(i-1,j-2) + T(i-1,j) for j = 2..2i-2.

1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 3, 1, 1, 1, 2, 5, 6, 8, 6, 5, 2, 1, 1, 2, 6, 8, 13, 12, 13, 8, 6, 2, 1, 1, 3, 9, 16, 27, 33, 38, 33, 27, 16, 9, 3, 1, 1, 3, 10, 19, 36, 49, 65, 66, 65, 49, 36, 19, 10, 3, 1, 1, 4, 14, 32, 65, 104, 150, 180, 196, 180

Offset: 0

Clark Kimberling

T(n, k) is the number of strings s(0)..s(n) such that s(0) = 0, s(n) = n-k, |s(i) - s(i-1)| <= 1 if i is even, |s(i) - s(i-1)| = 1 if i is odd.

			First 5 rows:
  1
  1  0  1
  1  1  2  1  1
  1  1  3  2  3  1  1
  1  2  5  6  8  6  5  2  1

Cf. A026537, A026538, A026539, A026540, A026541, A026545, A026546, A026547, A026548, A026549, A026550, A027267, A027268, A027269, A027270, A027271, A352972.

Cf. A026519, A026527, A026552, A026584, A027926.

z = 12; t[n_, 0] := 1; t[n_, k_] := 1 /; k == 2 n; t[n_, 1] := Floor[n/2];
t[n_, k_] := Floor[n/2] /; k == 2 n - 1; t[n_, k_] := t[n, k] =
If[EvenQ[n], t[n - 1, k - 2] + t[n - 1, k - 1] + t[n - 1, k], t[n - 1, k -
2] + t[n - 1, k]]; u = Table[t[n, k], {n, 0, z}, {k, 0, 2 n}];
TableForm[u]   (* A026536 array *)
v = Flatten[u] (* A026536 sequence *)

@cached_function
def T(n, k):
    if k < 0 or n < 0: return 0
    elif k == 0 or k == 2*n: return 1
    elif k == 1 or k == 2*n-1: return n//2
    elif n % 2 == 1: return T(n-1, k-2) + T(n-1, k)
    return T(n-1, k-2) + T(n-1, k-1) + T(n-1, k) # Peter Luschny, Oct 13 2019

Updated by Clark Kimberling, Aug 28 2014

Offset changed to 0 by Peter Luschny, Oct 10 2019

A026548 a(n) = T(n,0) + T(n,1) + ... + T(n,n), T given by A026536.

Original entry on oeis.org

1, 1, 4, 7, 22, 42, 127, 249, 746, 1476, 4414, 8766, 26215, 52158, 156041, 310799, 930194, 1854072, 5550976, 11070000, 33152042, 66139316, 198115526, 395368914, 1184511095, 2364457980, 7084871668, 14145343660, 42390336619

Offset: 0

Clark Kimberling

nonn

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

Cf. A026536, A026550, A027271.

T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1,  Floor[n/2],
If[EvenQ[n], T[n-1,k-2] +T[n-1,k-1] +T[n-1,k], T[n-1, k-2] +T[n-1,k]]]];
Table[Sum[T[n, k], {k,0,n}], {n,0,40}] (* G. C. Greubel, Apr 12 2022 *)

@CachedFunction
def T(n, k): # A026536
    if k == 0 or k == 2*n: return 1
    elif k == 1 or k == 2*n-1: return n//2
    elif n % 2 == 1: return T(n-1, k-2) + T(n-1, k)
    return T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
def A026548(n): return sum(T(n,k) for k in (0..n))
[A026548(n) for n in (0..40)] # G. C. Greubel, Apr 12 2022

a(n) = Sum_{k=0..n} A026536(n, k).

Missing a(0)=1 inserted by Sean A. Irvine, Oct 06 2019

A352972 a(n) = Sum_{j=0..2*n} Sum_{k=0..j} A026536(j, k).

Original entry on oeis.org

1, 6, 35, 204, 1199, 7089, 42070, 250269, 1491262, 8896310, 53118352, 317373194, 1897253203, 11346582851, 67882263130, 406231442387, 2431626954934, 14558306758418, 87177151134954, 522110098886882, 3127380060424476, 18734897945679836, 112245303177542790, 672552484035697364, 4030148584900522009

Offset: 0

G. C. Greubel, Apr 12 2022

nonn

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

Cf. A026536, A026550.

T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[n/2], If[EvenQ[n], T[n-1, k-2] +T[n-1, k-1] +T[n-1, k], T[n-1, k-2] +T[n-1, k]] ]];
A352972[n_]:= A352972[n]= Sum[T[j,k], {j,0,2*n}, {k,0,j}];
Table[A352972[n], {n,0,40}]

@CachedFunction
def T(n, k): # A026536
    if k == 0 or k == 2*n: return 1
    elif k == 1 or k == 2*n-1: return n//2
    elif n % 2 == 1: return T(n-1, k-2) + T(n-1, k)
    return T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
def A352972(n): return sum(sum(T(j,k) for k in (0..j)) for j in (0..2*n))
[A352972(n) for n in (3..40)]

a(n) = Sum_{j=0..2*n} Sum_{k=0..j} A026536(j, k).

cp's OEIS Frontend

A026536 Irregular triangular array T read by rows: T(i,0 ) = T(i,2i) = 1 for i >= 0; T(i,1) = T(i,2i-1) = floor(i/2) for i >= 1; for even n >= 2, T(i,j) = T(i-1,j-2) + T(i-1,j-1) + T(i-1,j) for j = 2..2i-2, for odd n >= 3, T(i,j) = T(i-1,j-2) + T(i-1,j) for j = 2..2i-2.

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A026548 a(n) = T(n,0) + T(n,1) + ... + T(n,n), T given by A026536.

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A352972 a(n) = Sum_{j=0..2*n} Sum_{k=0..j} A026536(j, k).

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