A026536 -id:A026536 - cp's OEIS frontend

A027267 a(n) = self-convolution of row n of array T given by A026536.

1, 2, 8, 26, 196, 692, 5774, 21142, 180772, 675344, 5837908, 22087716, 192239854, 733698032, 6416509142, 24645099530, 216309089956, 834847581048, 7347943049432, 28467646552432, 251119894730596, 975892708569952, 8624336421678788, 33600628889991916, 297394187356638766

Offset: 0

Clark Kimberling

nonn

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

Cf. A026536.

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]*T[n,2*n-k], {k,0,2*n}], {n,0,40}] (* G. C. Greubel, Apr 12 2022 *)

@CachedFunction
def T(n, k): # A026536
    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)
def A027267(n): return sum(T(n,k)*T(n,2*n-k) for k in (0..2*n))
[A027267(n) for n in (0..40)] # G. C. Greubel, Apr 12 2022

a(n) = Sum_{k=0..2*n} A026536(n, k)*A026536(n, 2*n-k). - G. C. Greubel, Apr 12 2022

More terms from Sean A. Irvine, Oct 26 2019

A026550 a(n) = Sum_{i=0..n} Sum_{j=0..i} T(i,j), T given by A026536.

Original entry on oeis.org

1, 2, 6, 13, 35, 77, 204, 453, 1199, 2675, 7089, 15855, 42070, 94228, 250269, 561068, 1491262, 3345334, 8896310, 19966310, 53118352, 119257668, 317373194, 712742108, 1897253203, 4261711183, 11346582851, 25491926511, 67882263130

Offset: 0

Clark Kimberling

nonn

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

Cf. A026548.

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]] ]];
A026550[n_]:= A026550[n]= Sum[T[j, k], {j,0,n}, {k,0,j}];
Table[A026550[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 A026550(n): return sum(sum(T(j,k) for k in (0..j)) for j in (0..n))
[A026550(n) for n in (0..40)] # G. C. Greubel, Apr 12 2022

a(n) = Sum_{j=0..n} Sum_{k=0..j} A026548(j, k).

A027268 a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A026536.

Original entry on oeis.org

0, 6, 20, 180, 644, 5502, 20292, 174456, 654632, 5673140, 21528000, 187675644, 717800628, 6284986554, 24178479500, 212408191568, 820811282352, 7229648901024, 28037230854096, 247468885359240, 962488105227160, 8510025522045036, 33177800527098040, 293772371437293720

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..500

Cf. A026536, A027267, A027269.

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]*T[n,k+1], {k,0,2*n-1}], {n,40}] (* G. C. Greubel, Apr 12 2022 *)

@CachedFunction
def T(n, k): # A026536
    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)
def A027268(n): return sum(T(n,k)*T(n,k+1) for k in (0..2*n-1))
[A027268(n) for n in (1..40)] # G. C. Greubel, Apr 12 2022

a(n) = Sum_{k=0..2n-1} A026536(n,k) * A026536(n,k+1)

More terms from Sean A. Irvine, Oct 26 2019

A027269 a(n) = Sum_{k=0..2n-2} T(n,k) * T(n,k+2), with T given by A026536.

Original entry on oeis.org

1, 5, 19, 150, 561, 4797, 18089, 156900, 596674, 5205950, 19932353, 174609162, 672106267, 5906040623, 22829936683, 201114700568, 780077588440, 6885880226784, 26784015828458, 236826459554380, 923352937530146, 8175978023317170, 31940549289135429, 283166067626865540

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..500

Cf. A026536, A027267, A027268, A027270.

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]*T[n,k+2], {k,0,2*n-2}], {n,40}] (* G. C. Greubel, Apr 12 2022 *)

@CachedFunction
def T(n, k): # A026536
    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)
def A027269(n): return sum(T(n,k)*T(n,k+2) for k in (0..2*n-2))
[A027269(n) for n in (1..40)] # G. C. Greubel, Apr 12 2022

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

More terms from Sean A. Irvine, Oct 26 2019

a(1) = 1 prepended by G. C. Greubel, Apr 12 2022

A026537 a(n) = T(n,n), T given by A026536. Also a(n) = number of integer strings s(0), ..., s(n), counted by T, such that s(n)=0.

Original entry on oeis.org

1, 0, 2, 2, 8, 12, 38, 66, 196, 360, 1052, 1980, 5774, 11004, 32146, 61726, 180772, 348912, 1024256, 1984608, 5837908, 11346280, 33433996, 65143716, 192239854, 375351288, 1109049320, 2169299288, 6416509142, 12569973108

Offset: 0

Clark Kimberling

nonn

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

Cf. A026521, A026536.

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[T[n, n], {n, 0, 35}] (* G. C. Greubel, Apr 10 2022 *)

@cached_function
def T(n, k): # A026536
    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)
def A026537(n): return T(n,n)
[A026537(n) for n in (0..35)] # G. C. Greubel, Apr 10 2022

a(n) = A026536(n, n).

a(n) = 2 * A026521(n-1).

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

A027270 a(n) = Sum_{k=0..2n-3} T(n,k) * T(n,k+3), with T given by A026536.

Original entry on oeis.org

2, 10, 104, 420, 3786, 14826, 131264, 510576, 4508580, 17523506, 154773696, 602175444, 5323519838, 20744201142, 183586707648, 716553432640, 6348284151024, 24816637181076, 220081449149440, 861581808936200, 7647723960962932, 29978812970646870, 266322435212031984

Offset: 2

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 2..500

Cf. A026536, A027267, A027268, A027269.

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]*T[n,k+3], {k,0,2*n-3}], {n,2,40}] (* 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 A027270(n): return sum(T(n,k)*T(n,k+3) for k in (0..2*n-3))
[A027270(n) for n in (2..40)] # G. C. Greubel, Apr 12 2022

a(n) = Sum_{k=0..2n-3} A026536(n,k) * A026536(n,k+3).

More terms from Sean A. Irvine, Oct 26 2019

a(2) = 2 prepended by G. C. Greubel, Apr 12 2022

A027271 a(n) = Sum_{k=0..2n} (k+1)*T(n,k), where T is given by A026536.

Original entry on oeis.org

1, 4, 18, 48, 180, 432, 1512, 3456, 11664, 25920, 85536, 186624, 606528, 1306368, 4199040, 8957952, 28553472, 60466176, 191476224, 403107840, 1269789696, 2660511744, 8344332288, 17414258688, 54419558400, 113192681472, 352638738432, 731398864896, 2272560758784

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. A026536, A053469, A199299 (bisection).

[Round(6^(n/2)*( 3*((n+1) mod 2) + Sqrt(6)*(n mod 2) )*(n+1)/3): n in [0..40]]; // G. C. Greubel, Apr 12 2022

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]] ]];
A027271[n_]:= A027271[n]= Sum[(k+1)*T[n,k], {k,0,2*n}];
Table[A027271[n], {n,0,40}] (* G. C. Greubel, Apr 12 2022 *)

A027271(n)=my(b(n)=if(!bittest(n,0),n\2*6^(n\2-1)));4*b(n+1)+b(n+2)+6*b(n) \\ could be made more efficient and explicit by simplifying the formula for n even and for n odd separately. - M. F. Hasler, Sep 29 2012

[6^(n/2)*( 3*((n+1)%2) + sqrt(6)*(n%2) )*(n+1)/3 for n in (0..40)] # G. C. Greubel, Apr 12 2022

From Paul Barry, Mar 03 2004: (Start)
G.f.: (1+4*x+6*x^2)/(1-6*x^2)^2 = (d/dx)((1+3*x)/(1-6*x^2)).
a(n) = 6^(n/2)*((3-sqrt(6))*(-1)^n + (3+sqrt(6)))*(n+1)/6. (End)
a(n) = 4*b(n) + b(n+1) + 6*b(n-1) with b(n)= 0, 1, 0, 12, 0, 108, 0, 864, ... (aerated A053469). - R. J. Mathar, Sep 29 2012
E.g.f.: (1 + 2*x)*cosh(sqrt(6)*x) + sqrt(2/3)*(1 + 3*x)*sinh(sqrt(6)*x). - Stefano Spezia, May 07 2023

A026538 a(n) = T(n,n-1), T given by A026536. Also a(n) = number of integer strings s(0), ..., s(n), counted by T, such that s(n) = 1.

Original entry on oeis.org

1, 1, 3, 6, 13, 33, 65, 180, 346, 990, 1897, 5502, 10571, 30863, 59523, 174456, 337672, 992304, 1926650, 5673140, 11043858, 32571858, 63548069, 187675644, 366849016, 1084649644, 2123604927, 6284986554, 12322549765, 36501029265

Offset: 1

Clark Kimberling

nonn

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

Cf. A026536.

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[T[n, n-1], {n, 35}] (* G. C. Greubel, Apr 10 2022 *)

@CachedFunction
def T(n, k): # A026536
    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)
def A026538(n): return T(n,n-1)
[A026538(n) for n in (1..35)] # G. C. Greubel, Apr 10 2022

a(n) = A026536(n, n-1).

A026539 a(n) = T(n,n-2), T given by A026536. Also a(n) = number of integer strings s(0), ..., s(n), counted by T, such that s(n) = 2.

Original entry on oeis.org

1, 1, 5, 8, 27, 49, 150, 284, 845, 1625, 4797, 9288, 27377, 53207, 156900, 305720, 902394, 1761882, 5205950, 10181720, 30114073, 58983859, 174609162, 342449340, 1014555607, 1992082339, 5906040623, 11608506392, 34438443075

Offset: 2

Clark Kimberling

nonn

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

Cf. A026536.

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[T[n,n-2], {n,2,35}] (* G. C. Greubel, Apr 10 2022 *)

@CachedFunction
def T(n, k): # A026536
    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)
def A026539(n): return T(n,n-2)
[A026539(n) for n in (2..35)] # G. C. Greubel, Apr 10 2022

a(n) = A026536(n, n-2).

cp's OEIS Frontend

A027267 a(n) = self-convolution of row n of array T given by A026536.

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A026550 a(n) = Sum_{i=0..n} Sum_{j=0..i} T(i,j), T given by A026536.

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A027268 a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A026536.

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A027269 a(n) = Sum_{k=0..2n-2} T(n,k) * T(n,k+2), with T given by A026536.

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A026537 a(n) = T(n,n), T given by A026536. Also a(n) = number of integer strings s(0), ..., s(n), counted by T, such that s(n)=0.

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

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A027270 a(n) = Sum_{k=0..2n-3} T(n,k) * T(n,k+3), with T given by A026536.

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Mathematica

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A027271 a(n) = Sum_{k=0..2n} (k+1)*T(n,k), where T is given by A026536.

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