A026584 -id:A026584 - cp's OEIS frontend

A026596 Row sums of A026584.

1, 1, 4, 8, 23, 54, 143, 354, 914, 2306, 5907, 15012, 38368, 97804, 249865, 637834, 1629729, 4163398, 10640753, 27196246, 69526562, 177757762, 454541197, 1162403180, 2972953385, 7604223184, 19451741733, 49761433640, 127308417226

Offset: 0

Clark Kimberling

nonn

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

Cf. A026584, A026585, A026587, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026597, A026598, A026599, A027282, A027283, A027284, A027285, A027286.

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+k], T[n-1, k-2] + T[n-1, k], T[n-1, k-2] + T[n-1, k-1] + T[n-1, k] ]]]; (* T = A026584 *)
a[n_]:=a[n]= Sum[T[n,k], {k,0,n}];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Dec 13 2021 *)

@CachedFunction
def T(n, k):  # T = A026584
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n//2)
    else: return T(n-1, k-2) + T(n-1, k) if ((n+k)%2==0) else T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
@CachedFunction
def A026596(n): return sum( T(n, j) for j in (0..n) )
[A026596(n) for n in (0..40)] # G. C. Greubel, Dec 13 2021

a(n) = Sum_{k=0..n} A026584(n, k).

Conjecture: n*a(n) -3*(n-1)*a(n-1) -(5*n-6)*a(n-2) +3*(5*n-13)*a(n-3) +2*(4*n-9)*a(n-4) -8*(2*n-9)*a(n-5) = 0. - R. J. Mathar, Jun 23 2013

A026598 a(n) = Sum_{i=0..n} Sum_{j=0..i} T(i,j), T given by A026584.

Original entry on oeis.org

1, 2, 6, 14, 37, 91, 234, 588, 1502, 3808, 9715, 24727, 63095, 160899, 410764, 1048598, 2678327, 6841725, 17482478, 44678724, 114205286, 291963048, 746504245, 1908907425, 4881860810, 12486083994, 31937825727, 81699259367

Offset: 0

Clark Kimberling

nonn

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

Cf. A026584, A026585, A026587, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026596, A026597, A026599, A027282, A027283, A027284, A027285, A027286.

T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2*n, 1, If[k==1 || k==2*n - 1, Floor[n/2], If[EvenQ[n+k], T[n-1, k-2] + T[n-1, k], T[n-1, k-2] + T[n-1, k-1] + T[n-1, k] ]]]];
a[n_]:= a[n]= Block[{$RecursionLimit = Infinity}, Sum[T[i,j], {i,0,n}, {j,0,i}]];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Dec 15 2021 *)

@CachedFunction
def T(n, k):  # T = A026584
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n//2)
    else: return T(n-1, k-2) + T(n-1, k) if ((n+k)%2==0) else T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
@CachedFunction
def A026598(n): return sum(sum(T(i,j) for j in (0..i)) for i in (0..n))
[A026598(n) for n in (0..40)] # G. C. Greubel, Dec 15 2021

a(n) = Sum_{i=0..n} Sum_{j=0..i} A026584(i, j).

Conjecture: n*a(n) - (4*n-3)*a(n-1) - (2*n-3)*a(n-2) + 5*(4*n-9)*a(n-3) - 7*(n-3)*a(n-4) - 6*(4*n-15)*a(n-5) + 8*(2*n-9)*a(n-6) = 0. - R. J. Mathar, Jun 23 2013

A027286 a(n) = Sum_{k=0..2n} (k+1) * A026584(n, k).

Original entry on oeis.org

1, 4, 18, 56, 190, 564, 1722, 4976, 14454, 40940, 115698, 322728, 896558, 2471588, 6786090, 18537184, 50459366, 136844892, 370030434, 997705240, 2683514526, 7201203988, 19284880794, 51546789456, 137541880150, 366412976332

Offset: 0

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,7,-8,-16).

Cf. A000032, A000045, A006131, A026584, A072265.

I:=[1,4,18,56]; [n le 4 select I[n] else 2*Self(n-1) +7*Self(n-2) -8*Self(n-3) -16*Self(n-4): n in [1..31]]; // G. C. Greubel, Dec 12 2021

LinearRecurrence[{2,7,-8,-16},{1,4,18,56}, 30] (* G. C. Greubel, Dec 12 2021 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -16,-8,7,2]^n*[1;4;18;56])[1,1] \\ Charles R Greathouse IV, Oct 21 2022

[2^(n-1)*(n+1)*(14*lucas_number2(n+2, 1/2, -1) + 5*lucas_number2(n+1, 1/2, -1))/17 for n in (0..30)] # G. C. Greubel, Dec 12 2021

G.f.: (1+2*x+3*x^2)/(1-x-4*x^2)^2.
From G. C. Greubel, Dec 12 2021: (Start)
a(n) = 2^(n-3)*( -6*F(n+1, 1/2) + Sum_{j=0..n} F(n-j+1, 1/2)*( 14*F(j+1, 1/2) + 5*F(j, 1/2) ), where F(n, x) are the Fibonacci polynomials.
a(n) = (2^(n-1)/17)*(n+1)*( 14*L(n+2, 1/2) + 5*L(n+1, 1/2) ), where L(n, x) are the Lucas polynomials.
a(n) = 2*a(n-1) + 7*a(n-2) - 8*a(n-3) - 16*a(n-4). (End)

A027282 a(n) = self-convolution of row n of array T given by A026584.

Original entry on oeis.org

1, 2, 8, 40, 222, 1296, 7770, 47324, 291260, 1806220, 11266718, 70609316, 444231564, 2803975860, 17748069294, 112609964308, 716010467122, 4561107325336, 29103104031990, 185973253609716, 1189979068401564, 7623432519587692, 48891854980251090, 313874287333373820

Offset: 0

Clark Kimberling

nonn

Bisection of A026585.

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

Cf. A026584, A026585, A026587, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026596, A026597, A026598, A026599, A027283, A027284, A027285, A027286.

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+k], T[n-1, k-2] + T[n-1, k], T[n-1, k-2] + T[n-1, k-1] + T[n-1, k] ]]]; (* T = A026584 *)
a[n_]:= a[n]= Sum[T[n, k]*T[n, 2*n-k], {k,0,2*n}];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Dec 15 2021 *)

@CachedFunction
def T(n, k):  # T = A026584
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n//2)
    else: return T(n-1, k-2) + T(n-1, k) if ((n+k)%2==0) else T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
@CachedFunction
def A027282(n): return sum(T(n,j)*T(n, 2*n-j) for j in (0..2*n))
[A027282(n) for n in (0..40)] # G. C. Greubel, Dec 15 2021

a(n) = Sum_{k=0..2*n} A026584(n, k)*A026584(n, 2*n-k).

More terms from Sean A. Irvine, Oct 26 2019

A027283 a(n) = Sum_{k=0..2n-1} T(n,k) T(n,k+1), with T given by A026584.

Original entry on oeis.org

0, 6, 26, 206, 1100, 7314, 42920, 274010, 1677332, 10616070, 66290046, 419754586, 2648500908, 16818685050, 106781976774, 680250643910, 4337083126232, 27709045093274, 177213890858938, 1135003956744310, 7276652578220372, 46702733068082702, 300013046145979184

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A026584, A026585, A026587, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026596, A026597, A026598, A026599, A027282, A027284, A027285, A027286.

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+k], T[n-1, k-2] + T[n-1, k], T[n-1, k-2] + T[n-1, k-1] + T[n-1, k] ]]]; (* T = A026584 *)
a[n_]:= a[n]= Sum[T[n, k]*T[n, k+1], {k, 0, 2*n-1}];
Table[a[n], {n, 1, 40}] (* G. C. Greubel, Dec 15 2021 *)

@CachedFunction
def T(n, k):  # T = A026584
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n//2)
    else: return T(n-1, k-2) + T(n-1, k) if ((n+k)%2==0) else T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
@CachedFunction
def A027283(n): return sum(T(n,j)*T(n, j+1) for j in (0..2*n-1))
[A027283(n) for n in (1..40)] # G. C. Greubel, Dec 15 2021

a(n) = Sum_{k=0..2*n-1} A026584(n,k) * A026584(n,k+1).

More terms from Sean A. Irvine, Oct 26 2019

A027284 a(n) = Sum_{k=0..2n-2} T(n,k) T(n,k+2), with T given by A026584.

Original entry on oeis.org

5, 28, 167, 1024, 6359, 39759, 249699, 1573524, 9943905, 62994733, 399936573, 2543992514, 16210331727, 103453402718, 661164765879, 4230874777682, 27105456280491, 173838468040879, 1115987495619427, 7170725839251598, 46113396476943241, 296773029762031990

Offset: 2

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 2..1000

Cf. A026584, A026585, A026587, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026596, A026597, A026598, A026599, A027282, A027283, A027285, A027286.

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+k], T[n-1, k-2] + T[n-1, k], T[n-1, k-2] + T[n-1, k-1] + T[n-1, k] ]]]; (* T = A026584 *)
a[n_]:= a[n]= Sum[T[n, k]*T[n, k+2], {k, 0, 2*n-2}];
Table[a[n], {n, 2, 40}] (* G. C. Greubel, Dec 15 2021 *)

@CachedFunction
def T(n, k):  # T = A026584
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n//2)
    else: return T(n-1, k-2) + T(n-1, k) if ((n+k)%2==0) else T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
@CachedFunction
def A027284(n): return sum(T(n,j)*T(n, j+2) for j in (0..2*n-2))
[A027284(n) for n in (2..40)] # G. C. Greubel, Dec 15 2021

a(n) = Sum_{k=0..2*n-2} A026584(n,k) * A026584(n,k+2).

More terms from Sean A. Irvine, Oct 26 2019

A027285 a(n) = Sum_{k=0..2n-3} T(n,k) T(n,k+3), with T given by A026584.

Original entry on oeis.org

12, 116, 682, 4908, 30272, 201648, 1273286, 8275894, 52783298, 340392020, 2180905198, 14035736838, 90149817980, 580197442656, 3732734480794, 24041345351898, 154874693823022, 998441294531516, 6439238635990250, 41552345665859196, 268252644944872486

Offset: 3

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 3..1000

Cf. A026584, A026585, A026587, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026596, A026597, A026598, A026599, A027282, A027283, A027284, A027286.

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+k], T[n-1, k-2] + T[n-1, k], T[n-1, k-2] + T[n-1, k-1] + T[n-1, k] ]]]; (* T = A026584 *)
a[n_]:= a[n]= Sum[T[n, k]*T[n, k+3], {k, 0, 2*n-3}];
Table[a[n], {n, 3, 40}] (* G. C. Greubel, Dec 15 2021 *)

@CachedFunction
def T(n, k):  # T = A026584
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n//2)
    else: return T(n-1, k-2) + T(n-1, k) if ((n+k)%2==0) else T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
@CachedFunction
def A027285(n): return sum(T(n,j)*T(n, j+3) for j in (0..2*n-3))
[A027285(n) for n in (3..40)] # G. C. Greubel, Dec 15 2021

a(n) = Sum_{k=0..2*n-3} A026584(n,k) * A026584(n,k+3).

More terms from Sean A. Irvine, Oct 26 2019

A026597 Expansion of (1+x)/(1-x-4*x^2).

Original entry on oeis.org

1, 2, 6, 14, 38, 94, 246, 622, 1606, 4094, 10518, 26894, 68966, 176542, 452406, 1158574, 2968198, 7602494, 19475286, 49885262, 127786406, 327327454, 838473078, 2147782894, 5501675206, 14092806782, 36099507606, 92470734734

Offset: 0

Clark Kimberling

This sequence can generated by the following formula: a(n) = a(n-1) + 4*a(n-2) when n > 2; a[1] = 1, a[2] = 2. - Alex Vinokur (alexvn(AT)barak-online.net), Oct 21 2004
An elephant sequence, see A175654 and A175655. For the corner squares just one A[5] vector, with decimal value 325, leads to the sequence given above. For the central square this vector leads to a companion sequence that is 4 times this very same sequence with n >= -1. - Johannes W. Meijer, Aug 15 2010
Equals INVERTi transform of A180168. - Gary W. Adamson, Aug 14 2010
Start with a single cell at coordinates (0, 0), then iteratively subdivide the grid into 2 X 2 cells and remove the cells that have one '1' in their modulo 3 coordinates. a(n) is the number of cells after n iterations. Cell configuration converges to a fractal with approximate dimension 1.357. - Peter Karpov, Apr 20 2017
Also, the number of walks of length n starting at vertex 1 in the graph with 4 vertices and edges {{0,1}, {0,2}, {0,3}, {1,2}, {2,3}}. - Sean A. Irvine, Jun 02 2025

Nathaniel Johnston, Table of n, a(n) for n = 0..500
Sean A. Irvine, Walks on Graphs.
Peter Karpov, InvMem, Item 26
Peter Karpov, Illustration of initial terms (n = 1..8)
Index entries for linear recurrences with constant coefficients, signature (1,4).

Cf. A006131, A006138, A007482, A026581, A026584, A122950, A175654, A175655, A180168.

[n le 2 select n else Self(n-1) + 4*Self(n-2): n in [1..41]]; // G. C. Greubel, Dec 08 2021

LinearRecurrence[{1,4},{1,2},40] (* Harvey P. Dale, Nov 28 2011 *)

[(2*i)^n*( chebyshev_U(n, -i/4) - (i/2)*chebyshev_U(n-1, -i/4) ) for n in (0..40)] # G. C. Greubel, Dec 08 2021

G.f.: (1+x)/(1-x-4*x^2).
a(n) = T(n,0) + T(n,1) + ... + T(n,2*n), T given by A026584.
a(n) = Sum_{k=0..n} binomial(floor((2*n-k-1)/2), n-k)*2^k. - Paul Barry, Feb 11 2005
a(n) = A006131(n) + A006131(n-1), n >= 1. - R. J. Mathar, Oct 20 2006
a(n) = Sum_{k=0..n} binomial(floor((2*n-k)/2),n-k)*4^floor(k/2). - Paul Barry, Feb 02 2007
Inverse binomial transform of A007482: (1, 3, 11, 39, 139, 495, ...). - Gary W. Adamson, Dec 04 2007
a(n) = Sum_{k=0..n+1} A122950(n+1,k)*3^(n+1-k). - Philippe Deléham, Jan 04 2008
a(n) = (1/2 + 3*sqrt(17)/34)*(1/2 + sqrt(17)/2)^n + (1/2 - 3*sqrt(17)/34)*(1/2 - sqrt(17)/2)^n. - Antonio Alberto Olivares, Jun 07 2011
a(n) = (2*i)^n*( chebyshevU(n, -i/4) - (i/2)*chebyshevU(n-1, -i/4) ). - G. C. Greubel, Dec 08 2021
E.g.f.: exp(x/2)*(17*cosh(sqrt(17)*x/2) + 3*sqrt(17)*sinh(sqrt(17)*x/2))/17. - Stefano Spezia, Jan 31 2023

Better name from Ralf Stephan, Jul 14 2013

A026552 Irregular triangular array T read by rows: T(n, 0) = T(n, 2n) = 1, T(n, 1) = T(n, 2n-1) = floor(n/2 + 1), for even n >= 2, T(n, k) = T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), otherwise T(n, k) = T(n-1, k-2) + T(n-1, k).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 2, 4, 4, 4, 2, 1, 1, 3, 7, 10, 12, 10, 7, 3, 1, 1, 3, 8, 13, 19, 20, 19, 13, 8, 3, 1, 1, 4, 12, 24, 40, 52, 58, 52, 40, 24, 12, 4, 1, 1, 4, 13, 28, 52, 76, 98, 104, 98, 76, 52, 28, 13, 4, 1, 1, 5, 18, 45, 93, 156, 226, 278

Offset: 0

Clark Kimberling

T(n, k) = number of integer strings s(0)..s(n) such that s(0) = 0, s(n) = n-k, |s(i)-s(i-1)|<=1 if i is even or i = 1, |s(i)-s(i-1)| = 1 if i is odd and i >= 3.

			First 5 rows:
  1;
  1, 1, 1;
  1, 2, 3,  2,  1;
  1, 2, 4,  4,  4,  2,  1;
  1, 3, 7, 10, 12, 10,  7,  3,  1;

Clark Kimberling, Table of n, a(n) for n = 0..10200 [Offset changed to 0 by _Georg Fischer_, Mar 01 2022]
Index entries for triangles and arrays related to Pascal's triangle

Cf. A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026563, A026566, A026567, A027272, A027273, A027274, A027275, A027276.
Cf. A026519, A026536, A026568, A026584, A027926.
Cf. A026565.

z = 12; t[n_, 0] := 1; t[n_, k_] := 1 /; k == 2 n; t[n_, 1] := Floor[n/2 + 1]; t[n_, k_] := Floor[n/2 + 1] /; 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] (* A026552 array *)
v = Flatten[u] (* A026552 sequence *)

@CachedFunction
def T(n,k): # T = A026552
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+2)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-2)
flatten([[T(n,k) for k in (0..2*n)] for n in (0..10)]) # G. C. Greubel, Dec 17 2021

Sum_{k=0..2*n} T(n,k) = A026565(n). - G. C. Greubel, Dec 17 2021

Updated by Clark Kimberling, Aug 28 2014

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.

Original entry on oeis.org

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

cp's OEIS Frontend

A026596 Row sums of A026584.

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A026598 a(n) = Sum_{i=0..n} Sum_{j=0..i} T(i,j), T given by A026584.

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A027286 a(n) = Sum_{k=0..2n} (k+1) * A026584(n, k).

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A027282 a(n) = self-convolution of row n of array T given by A026584.

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A027283 a(n) = Sum_{k=0..2*n-1} T(n,k) * T(n,k+1), with T given by A026584.

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A027284 a(n) = Sum_{k=0..2*n-2} T(n,k) * T(n,k+2), with T given by A026584.

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A027285 a(n) = Sum_{k=0..2*n-3} T(n,k) * T(n,k+3), with T given by A026584.

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A026597 Expansion of (1+x)/(1-x-4*x^2).

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A027283 a(n) = Sum_{k=0..2n-1} T(n,k) T(n,k+1), with T given by A026584.

A027284 a(n) = Sum_{k=0..2n-2} T(n,k) T(n,k+2), with T given by A026584.

A027285 a(n) = Sum_{k=0..2n-3} T(n,k) T(n,k+3), with T given by A026584.

A026552 Irregular triangular array T read by rows: T(n, 0) = T(n, 2n) = 1, T(n, 1) = T(n, 2n-1) = floor(n/2 + 1), for even n >= 2, T(n, k) = T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), otherwise T(n, k) = T(n-1, k-2) + T(n-1, k).