cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A026594 a(n) = T(2*n-1, n-2), where T is given by A026584.

Original entry on oeis.org

1, 2, 13, 42, 225, 802, 4235, 15478, 82425, 304156, 1634435, 6064389, 32819839, 122244344, 665162897, 2484851486, 13577768505, 50841782786, 278745377821, 1045763359942, 5749240499515, 21603797860416, 119040956286133, 447922312642212, 2472886893122590, 9315646385012666, 51514464212546865, 194255376492836212
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026584, A026585, A026587, A026589, A026590, A026591, A026592, A026593, A026595, A026596, A026597, A026598, A026599, A027282, A027283, A027284, A027285, A027286.

Programs

  • Mathematica
    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*)
Table[T[2*n-1, n-2], {n, 2, 40}]  (* G. C. Greubel, Dec 13 2021 *)
  • Sage
    @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)
[T(2*n-1, n-2) for n in (2..40)] # G. C. Greubel, Dec 13 2021

Formula

a(n) = A026584(2*n-1, n-2).

Extensions

Terms a(19) onward added by G. C. Greubel, Dec 13 2021

A026596 Row sums of A026584.

Original entry on oeis.org

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

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • Sage
    @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

Formula

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

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • Sage
    @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

Formula

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

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

Views

Author

Clark Kimberling

Keywords

Comments

Bisection of A026585.

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • Sage
    @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

Formula

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

Extensions

More terms from Sean A. Irvine, Oct 26 2019

A027283 a(n) = Sum_{k=0..2*n-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

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • Sage
    @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

Formula

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

Extensions

More terms from Sean A. Irvine, Oct 26 2019

A027284 a(n) = Sum_{k=0..2*n-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

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • Sage
    @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

Formula

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

Extensions

More terms from Sean A. Irvine, Oct 26 2019

A027285 a(n) = Sum_{k=0..2*n-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

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • Sage
    @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

Formula

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

Extensions

More terms from Sean A. Irvine, Oct 26 2019

A360309 a(n) = Sum_{k=0..floor(n/3)} binomial(n-1-2*k,n-3*k) * binomial(2*k,k).

Original entry on oeis.org

1, 0, 0, 2, 2, 2, 8, 14, 20, 46, 92, 158, 314, 630, 1176, 2274, 4498, 8674, 16804, 32990, 64358, 125414, 245832, 481674, 942912, 1850122, 3633220, 7133730, 14020694, 27578954, 54261912, 106819006, 210411028, 414619486, 817344908, 1611978734, 3180333830, 6276743430
Offset: 0

Views

Author

Seiichi Manyama, Feb 03 2023

Keywords

Crossrefs

Cf. A026585, A360310.
Cf. A360291, A360314.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(n-1-2*k, n-3*k)*binomial(2*k, k));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(1/sqrt(1-4*x^3/(1-x)))

Formula

G.f.: 1 / sqrt(1-4*x^3/(1-x)).
n*a(n) = 2*(n-1)*a(n-1) - (n-2)*a(n-2) + 2*(2*n-3)*a(n-3) - 2*(2*n-6)*a(n-4).
a(n) ~ 2^(n-1) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 18 2023

A377204 Expansion of 1/(1 - 4*x^2/(1-x))^(3/2).

Original entry on oeis.org

1, 0, 6, 6, 36, 66, 236, 546, 1626, 4106, 11388, 29646, 79838, 209718, 557328, 1465970, 3869448, 10166370, 26726080, 70092570, 183756378, 481048010, 1258494768, 3289100958, 8590288128, 22418099982, 58467588768, 152388145382, 396954437202, 1033452111702, 2689186662552
Offset: 0

Views

Author

Seiichi Manyama, Oct 20 2024

Keywords

Links

Crossrefs

Cf. A377197, A377213.
Cf. A026585, A377215.
Cf. A377186.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/(1 - 4*x^2/(1-x))^(3/2))); // Vincenzo Librandi, May 08 2025
  • Mathematica
    Table[Sum[(2*k+1)*Binomial[2*k,k]*Binomial[n-k-1,n-2*k],{k,0,Floor[n/2]}],{n,0,35}] (* Vincenzo Librandi, May 08 2025 *)
  • PARI
    a(n) = sum(k=0, n\2, (2*k+1)*binomial(2*k, k)*binomial(n-k-1, n-2*k));

Formula

a(n) = (2*(n-1)*a(n-1) + (3*n+6)*a(n-2) - 2*(2*n-3)*a(n-3))/n for n > 2.
a(n) = Sum_{k=0..floor(n/2)} (2*k+1) * binomial(2*k,k) * binomial(n-k-1,n-2*k).
a(n) ~ sqrt(n) * 2^(3*n - 1/2) / (17^(3/4) * sqrt(Pi) * (sqrt(17) - 1)^(n - 3/2)). - Vaclav Kotesovec, May 03 2025

A360310 a(n) = Sum_{k=0..floor(n/4)} binomial(n-1-3*k,n-4*k) * binomial(2*k,k).

Original entry on oeis.org

1, 0, 0, 0, 2, 2, 2, 2, 8, 14, 20, 26, 52, 98, 164, 250, 426, 762, 1328, 2194, 3682, 6366, 11072, 18878, 32038, 54906, 94860, 163226, 279634, 479806, 826776, 1425542, 2454020, 4223170, 7279164, 12560466, 21671314, 37381714, 64512676, 111414042
Offset: 0

Views

Author

Seiichi Manyama, Feb 03 2023

Keywords

Crossrefs

Cf. A026585, A360309.
Cf. A360292, A360315.

Programs

  • PARI
    a(n) = sum(k=0, n\4, binomial(n-1-3*k, n-4*k)*binomial(2*k, k));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(1/sqrt(1-4*x^4/(1-x)))

Formula

G.f.: 1 / sqrt(1-4*x^4/(1-x)).
n*a(n) = 2*(n-1)*a(n-1) - (n-2)*a(n-2) + 2*(2*n-4)*a(n-4) - 2*(2*n-7)*a(n-5).
Previous Showing 11-20 of 23 results. Next

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