A026524 -id:A026524 - cp's OEIS frontend

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

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

Clark Kimberling

nonn

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

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

Cf. A026552.

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 *)

@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

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

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

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

Clark Kimberling

nonn

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

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

Cf. A026552.

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 *)

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 */

@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

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

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

Clark Kimberling

nonn

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

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

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 *)

@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

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

A026532 Ratios of successive terms are 3, 2, 3, 2, 3, 2, 3, 2, ...

Original entry on oeis.org

1, 3, 6, 18, 36, 108, 216, 648, 1296, 3888, 7776, 23328, 46656, 139968, 279936, 839808, 1679616, 5038848, 10077696, 30233088, 60466176, 181398528, 362797056, 1088391168, 2176782336, 6530347008, 13060694016, 39182082048, 78364164096, 235092492288, 470184984576

Offset: 1

Clark Kimberling

Preface the series with a 1: (1, 1, 3, 6, 18, 36, ...); then the next term in the series = (1, 1, 3, 6, ...) dot (1, 2, 1, 2, ...). Example: 36 = (1, 1, 3, 6, 18) dot (1, 2, 1, 2, 1) = (1 + 2 + 3 + 12 + 18). - Gary W. Adamson, Apr 18 2009

Partial products of A176059. - Reinhard Zumkeller, Apr 04 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..700
Sean A. Irvine, Walks on Graphs.
José L. Ramírez, Bi-periodic incomplete Fibonacci sequences, Annales Mathematicae et Informaticae 42 (2013), 83-92. See the 1st column of Table 1 on p. 85.
Index entries for linear recurrences with constant coefficients, signature (0,6).

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.
Cf. A026534, A026549, A176059, A208131.
Cf. A038730, A038792, and A134511 for incomplete Fibonacci sequences, and A324242 for incomplete Lucas sequences.

a026532 n = a026532_list !! (n-1)
a026532_list = scanl (*) 1 $ a176059_list
-- Reinhard Zumkeller, Apr 04 2012

[(1/4)*(3-(-1)^n)*6^Floor(n/2) : n in [1..30]]; // Vincenzo Librandi, Jun 08 2011

FoldList[(2 + Boole[EvenQ@ #2]) #1 &, Range@ 28] (* or *)
CoefficientList[Series[x*(1+3x)/(1-6x^2), {x,0,31}], x] (* Michael De Vlieger, Aug 02 2017 *)
LinearRecurrence[{0,6},{1,3},30] (* Harvey P. Dale, Jul 11 2018 *)

a(n)=if(n%2,3,1)*6^(n\2) \\ Charles R Greathouse IV, Jul 02 2013

def a(n): return (3 if n%2 else 1)*6**(n//2)
print([a(n) for n in range(31)]) # Indranil Ghosh, Aug 02 2017

[(1/2)*6^((n-2)/2)*(3*(1+(-1)^n) + sqrt(6)*(1-(-1)^n)) for n in (1..30)] # G. C. Greubel, Dec 21 2021

a(n) = T(n, 0) + T(n, 1) + ... + T(n, 2n-2), T given by A026519.
From Benoit Cloitre, Nov 14 2003: (Start)
a(n) = (1/2)*(5+(-1)^n)*a(n-1) for n>1, a(1) = 1.
a(n) = (1/4)*(3-(-1)^n)*6^floor(n/2).  (End)
From Ralf Stephan, Feb 03 2004: (Start)
G.f.: x*(1+3*x)/(1-6*x^2).
a(n+2) = 6*a(n). (End)
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = (1/2)*6^((n-2)/2)*(3*(1+(-1)^n) + sqrt(6)*(1-(-1)^n)). - G. C. Greubel, Dec 21 2021
Sum_{n>=1} 1/a(n) = 8/5. - Amiram Eldar, Feb 13 2023

New definition from Ralf Stephan, Dec 01 2004

Offset changed from 0 to 1 by Vincenzo Librandi, Jun 08 2011

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

Clark Kimberling

nonn

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

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

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 *)

@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

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

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

Clark Kimberling

nonn

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

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

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 *)

@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

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

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

Clark Kimberling

nonn

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

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

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 *)

@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

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

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

Clark Kimberling

nonn

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

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

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 *)

@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

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

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

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,12,0,-36).

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

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

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 *)

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

[((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

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

Clark Kimberling

nonn

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

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

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 *)

@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

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

cp's OEIS Frontend

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

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

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

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A026532 Ratios of successive terms are 3, 2, 3, 2, 3, 2, 3, 2, ...

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Haskell

Magma

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PARI

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Sage

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

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Sage

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

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

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Mathematica