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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A026528 a(n) = T(2*n-1, n-1), T given by A026519.

Original entry on oeis.org

1, 2, 8, 28, 111, 436, 1763, 7176, 29521, 122182, 508595, 2126312, 8923136, 37563930, 158563368, 670893296, 2844444761, 12081753410, 51400091942, 218990735668, 934228356445, 3990177231742, 17060699906541, 73017457810032, 312785412844736, 1340988707637776, 5753539499846507
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026529, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.
Cf. A026552.

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+1)/2], If[EvenQ[n], T[n-1, k-2] + T[n-1, k], T[n-1, k-1] + T[n-1, k-2] + T[n-1, k] ]]]]; (* T = A026519 *)
a[n_]:= a[n]= Block[{$RecursionLimit = Infinity}, T[2*n-1, n-1] ];
Table[a[n], {n,40}] (* G. C. Greubel, Dec 20 2021 *)
  • Sage
    @CachedFunction
def T(n,k): # T = A026519
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+1)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
[T(2*n-1,n-1) for n in (1..40)] # G. C. Greubel, Dec 20 2021

Formula

a(n) = A026519(2*n-1, n-1).
a(n) = A026552(2*n-1, n-1).

Extensions

Terms a(20) onward added by G. C. Greubel, Dec 20 2021

A026529 a(n) = T(2*n-1, n-2), where T is given by A026519.

Original entry on oeis.org

1, 3, 13, 50, 205, 833, 3437, 14232, 59301, 248050, 1041469, 4385888, 18519306, 78376403, 332370925, 1412000824, 6008104249, 25601113893, 109229104313, 466577280830, 1995120743749, 8539562784258, 36583756253885, 156854365793800, 673028595199000, 2889847430222961, 12416501973954798, 53381063233213198
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.
Cf. A026552.

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+1)/2], If[EvenQ[n], T[n-1, k-2] + T[n-1, k], T[n-1, k-1] + T[n-1, k-2] + T[n-1, k] ]]]]; (* T = A026519 *)
a[n_]:= a[n]= Block[{$RecursionLimit = Infinity}, T[2*n-1, n-2] ];
Table[a[n], {n, 2, 40}] (* G. C. Greubel, Dec 20 2021 *)
  • Maxima
    a(n):=sum(binomial(n-1,i-1)*sum(binomial(j,n-j+2*i)*binomial(n,j),j,0,n),i,1,n/2); /* Vladimir Kruchinin, Jan 16 2015 */
  • Sage
    @CachedFunction
def T(n,k): # T = A026519
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+1)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
[T(2*n-1,n-2) for n in (2..40)] # G. C. Greubel, Dec 20 2021

Formula

a(n) = A026519(2*n-1, n-2).
a(n) = A026552(2*n-1, n-2).
a(n) = Sum_{i=0..floor(n/2)} C(n-1, i-1)*Sum_{j=0..n} C(j, n-j+2*i)*C(n, j). - Vladimir Kruchinin, Jan 16 2015

Extensions

Terms a(20) onward added by G. C. Greubel, Dec 20 2021

A026531 a(n) = T(n,0) + T(n,1) + ... + T(n,n), T given by A026519.

Original entry on oeis.org

1, 2, 4, 11, 22, 64, 127, 376, 746, 2222, 4414, 13180, 26215, 78373, 156041, 466840, 930194, 2784266, 5550976, 16620976, 33152042, 99291358, 198115526, 593484440, 1184511095, 3548969075, 7084871668, 21230215328, 42390336619
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026534, A027262, A027263, A027264, A027265, A027266.

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+1)/2], If[EvenQ[n], T[n-1, k-2] + T[n-1, k], T[n-1, k-1] + T[n - 1, k-2] + T[n-1, k] ]]]]; (* T = A026519 *)
a[n_]:= a[n]= Block[{$RecursionLimit = Infinity}, Sum[T[n, j], {j,0,n}] ];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Dec 20 2021 *)
  • Sage
    @CachedFunction
def T(n,k): # T = A026519
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+1)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
@CachedFunction
def a(n): return sum( T(n,k) for k in (0..n) )
[a(n) for n in (0..40)] # G. C. Greubel, Dec 20 2021

Formula

a(n) = Sum_{j=0..n} A026519(n, j).

A027262 a(n) = self-convolution of row n of array T given by A026519.

Original entry on oeis.org

1, 3, 8, 58, 196, 1608, 5774, 48924, 180772, 1553940, 5837908, 50618184, 192239854, 1676640462, 6416509142, 56201554888, 216309089956, 1900789437276, 7347943049432, 64734185205960, 251119894730596, 2216888144737508, 8624336421678788, 76265067399850848, 297394187356638766
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026532, A026533, A026534, A027263, A027264, A027265, A027266.

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+1)/2], If[EvenQ[n], T[n-1, k-2] + T[n-1, k], T[n-1, k-1] + T[n-1, k-2] + T[n-1, k] ]]]]; (* T = A026519 *)
a[n_]:= a[n]= Block[{$RecursionLimit = Infinity}, Sum[T[n,k]*T[n,2*n-k], {k,0,2*n}] ];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Dec 21 2021 *)
  • Sage
    @CachedFunction
def T(n,k): # T = A026519
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+1)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
@CachedFunction
def a(n): return sum( T(n,k)*T(n,2*n-k) for k in (0..2*n) )
[a(n) for n in (0..40)] # G. C. Greubel, Dec 22 2021

Formula

a(n) = Sum_{k=0..2*n} A026519(n, k)*A026519(n, 2*n-k).

Extensions

More terms from Sean A. Irvine, Oct 26 2019

A027263 a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A026519.

Original entry on oeis.org

2, 6, 52, 180, 1516, 5502, 46936, 174456, 1504432, 5673140, 49288856, 187675644, 1639174304, 6284986554, 55108565584, 212408191568, 1868067054968, 7229648901024, 63734526307552, 247468885359240, 2185849699156352, 8510025522045036, 75288454939134992, 293772371437293720
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026532, A026533, A026534, A027262, A027264, A027265, A027266.

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+1)/2], If[EvenQ[n], T[n-1, k-2] + T[n-1, k], T[n-1, k-1] + T[n-1, k-2] + T[n-1, k] ]]]]; (* T = A026519 *)
a[n_]:= a[n]= Block[{$RecursionLimit = Infinity}, Sum[T[n, k]*T[n, k+1], {k, 0, 2*n-1}] ];
Table[a[n], {n, 40}] (* G. C. Greubel, Dec 21 2021 *)
  • Sage
    @CachedFunction
def T(n,k): # T = A026519
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+1)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
@CachedFunction
def a(n): return sum( T(n,k)*T(n,k+1) for k in (0..2*n-1) )
[a(n) for n in (1..40)] # G. C. Greubel, Dec 21 2021

Formula

a(n) = Sum_{k=0..2n-1} A026519(n,k) * A026519(n,k+1).

Extensions

More terms from Sean A. Irvine, Oct 26 2019

A027264 a(n) = Sum_{k=0..2n-2} T(n,k) * T(n,k+2), with T given by A026519.

Original entry on oeis.org

5, 40, 150, 1279, 4797, 41462, 156900, 1365014, 5205950, 45501743, 174609162, 1531614109, 5906040623, 51952990090, 201114700568, 1773182087440, 6885880226784, 60825762159338, 236826459554380, 2095280066101886, 8175978023317170, 72432026278468535, 283166067626865540
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026532, A026533, A026534, A027262, A027263, A027265, A027266.

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+1)/2], If[EvenQ[n], T[n-1, k-2] + T[n-1, k], T[n-1, k-1] + T[n-1, k-2] + T[n-1, k] ]]]]; (* T = A026519 *)
a[n_] := a[n] = Block[{$RecursionLimit = Infinity}, Sum[T[n, k]*T[n, k+2], {k, 0, 2*n-2}] ];
Table[a[n], {n, 2, 40}] (* G. C. Greubel, Dec 21 2021 *)
  • Sage
    @CachedFunction
def T(n,k): # T = A026519
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+1)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
@CachedFunction
def a(n): return sum( T(n,k)*T(n,k+2) for k in (0..2*n-2) )
[a(n) for n in (2..40)] # G. C. Greubel, Dec 21 2021

Formula

a(n) = Sum_{k=0..2n-2} A026519(n,k) * A026519(n,k+2).

Extensions

More terms from Sean A. Irvine, Oct 26 2019

A027265 a(n) = Sum_{k=0..2n-3} T(n,k) * T(n,k+3), with T given by A026519.

Original entry on oeis.org

24, 104, 954, 3786, 33648, 131264, 1159844, 4508580, 39809076, 154773696, 1367463642, 5323519838, 47082494816, 183586707648, 1625447736120, 6348284151024, 56265306436584, 220081449149440, 1952476424575980, 7647723960962932, 67907006619888744, 266322435212031984
Offset: 3

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026532, A026533, A026534, A027262, A027263, A027264, A027266.

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+1)/2], If[EvenQ[n], T[n-1, k-2] + T[n-1, k], T[n-1, k-1] + T[n-1, k-2] + T[n-1, k] ]]]]; (* T = A026519 *)
a[n_] := a[n] = Block[{$RecursionLimit = Infinity}, Sum[T[n, k]*T[n, k+3], {k, 0, 2*n-3}] ];
Table[a[n], {n, 3, 40}] (* G. C. Greubel, Dec 21 2021 *)
  • Sage
    @CachedFunction
def T(n,k): # T = A026519
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+1)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
@CachedFunction
def a(n): return sum( T(n,k)*T(n,k+3) for k in (0..2*n-3) )
[a(n) for n in (3..40)] # G. C. Greubel, Dec 21 2021

Formula

a(n) = Sum_{k=0..2n-3} A026519(n,k) * A026519(n,k+3).

Extensions

More terms from Sean A. Irvine, Oct 26 2019

A027266 a(n) = Sum_{k=0..2n} (k+1) * A026519(n, k).

Original entry on oeis.org

1, 6, 18, 72, 180, 648, 1512, 5184, 11664, 38880, 85536, 279936, 606528, 1959552, 4199040, 13436928, 28553472, 90699264, 191476224, 604661760, 1269789696, 3990767616, 8344332288, 26121388032, 54419558400, 169789022208
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026532, A026533, A026534, A027262, A027263, A027264, A027265.

Programs

  • Magma
    I:=[1,6,18,72]; [n le 4 select I[n] else 12*(Self(n-2) - 3*Self(n-4)): n in [1..41]]; // G. C. Greubel, Dec 21 2021
  • Mathematica
    CoefficientList[Series[(1+6x+6x^2)/(1-6x^2)^2,{x,0,30}],x] (* or *) LinearRecurrence[{0,12,0,-36},{1,6,18,72},30] (* Harvey P. Dale, Jun 19 2015 *)
  • PARI
    a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -36,0,12,0]^n*[1;6;18;72])[1,1] \\ Charles R Greathouse IV, Oct 18 2022
  • Sage
    [((n+1)/2)*6^((n-1)/2)*( 3*(1-(-1)^n) + sqrt(6)*(1+(-1)^n) ) for n in (0..40)] # G. C. Greubel, Dec 21 2021

Formula

a(n) = Sum_{k=0..2n} (k+1) * A026519(n, k).
G.f.: (1+6*x+6*x^2)/(1-6*x^2)^2.
a(n) = 12*a(n-2) - 36*a(n-4), with a(0)=1, a(1)=6, a(2)=18, a(3)=72. - Harvey P. Dale, Jun 19 2015
a(n) = ((n+1)/2)*6^((n-1)/2)*( 3*(1-(-1)^n) + sqrt(6)*(1+(-1)^n) ). - G. C. Greubel, Dec 21 2021

A026533 a(n) = Sum_{i=0..n} Sum_{j=0..i} T(i,j), T given by A026519.

Original entry on oeis.org

1, 3, 7, 18, 40, 104, 231, 607, 1353, 3575, 7989, 21169, 47384, 125757, 281798, 748638, 1678832, 4463098, 10014074, 26635050, 59787092, 159078450, 357193976, 950678416, 2135189511, 5684158586, 12769030254, 33999245582
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026534, A027262, A027263, A027264, A027265, A027266.

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+1)/2], If[EvenQ[n], T[n-1, k-2] + T[n-1, k], T[n-1, k-1] + T[n-1, k-2] + T[n-1, k] ]]]]; (* T = A026519 *)
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 20 2021 *)
  • Sage
    @CachedFunction
def T(n,k): # T = A026519
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+1)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
@CachedFunction
def a(n): return sum(sum( T(i,j) for j in (0..i)) for i in (0..n) )
[a(n) for n in (0..40)] # G. C. Greubel, Dec 20 2021

Formula

a(n) = Sum_{i=0..n} Sum_{j=0..i} A026519(i,j).

A026552 Irregular triangular array T read by rows: T(n, 0) = T(n, 2*n) = 1, T(n, 1) = T(n, 2*n-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

Views

Author

Clark Kimberling

Keywords

Comments

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.

Examples

			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;

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)
  • Sage
    @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

Formula

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

Extensions

Updated by Clark Kimberling, Aug 28 2014
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