A026591 -id:A026591 - cp's OEIS frontend

A026584 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; and for i >= 2 and j = 2..2i-2, T(i,j) = T(i-1,j-2) + T(i-1,j-1) + T(i-1,j) if i+j is odd, and T(i,j) = T(i-1,j-2) + T(i-1,j) if i+j is even.

1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 1, 4, 2, 4, 1, 1, 1, 2, 5, 7, 8, 7, 5, 2, 1, 1, 2, 8, 9, 20, 14, 20, 9, 8, 2, 1, 1, 3, 9, 19, 28, 43, 40, 43, 28, 19, 9, 3, 1, 1, 3, 13, 22, 56, 62, 111, 86, 111, 62, 56, 22, 13, 3, 1, 1, 4, 14, 38, 69, 140, 167, 259, 222, 259, 167, 140, 69, 38, 14, 4, 1

Offset: 1

Clark Kimberling

Row sums are in A026597. - Philippe Deléham, Oct 16 2006

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 s(i-1) odd, |s(i)-s(i-1)| = 1 if s(i-1) is even, for i = 1..n.

			First 5 rows:
  1
  1  0  1
  1  1  2  1  1
  1  1  4  2  4  1  1
  1  2  5  7  8  7  5  2  1

Clark Kimberling, Rows 0..100, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A026519, A026536, A026552, A026568, A027926.

Cf. A026585, A026587, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026596, A026597, A026598, A026599.

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 + k], t[n - 1, k - 2] + t[n - 1, k], t[n - 1, k - 2] + t[n - 1, k - 1] + t[n - 1, k]]; u = Table[t[n, k], {n, 0, z}, {k, 0, 2 n}];
TableForm[u]   (* A026584 array *)
v = Flatten[u] (* A026584 sequence *)

@CachedFunction
def T(n,k):
    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)
flatten([[T(n,k) for k in (0..2*n)] for n in (0..12)]) # G. C. Greubel, Dec 11 2021

T(n, k) = T(n-1, k-2) + T(n-1, k) if ( (n+k) mod 2 ) = 0, otherwise T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), where T(n, 0) = T(n, 2*n) = 1, T(n, 1) = T(n, 2*n-1) = floor(n/2).

Updated by Clark Kimberling, Aug 29 2014

A026585 a(n) = T(n,n), T given by A026584. Also a(n) is the 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, 14, 40, 86, 222, 518, 1296, 3130, 7770, 19066, 47324, 117094, 291260, 724302, 1806220, 4507230, 11266718, 28188070, 70609316, 177023466, 444231564, 1115639586, 2803975860, 7052132546, 17748069294, 44693162266

Offset: 0

Clark Kimberling

nonn

The signed sequence 1,0,2,-2,8,-14,... is the inverse binomial transform of A026569. - Paul Barry, Sep 09 2004

Hankel transform of a(n) is 2^n. Hankel transform of a(n+1) is {0, -4, 0, 16, 0, -64, 0, 256, 0, ...} or -2^(n+1)*[x^n](x/(1+x^2)). Hankel transform of a(n+2) is 2^(n+1)*A109613(n+1). - Paul Barry, Mar 23 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

Cf. A026584, A026587, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026596, A026597, A026598, A026599, A027282, A027283, A027284, A027285, A027286.

Cf. A000984, A026569, A109613.

[(&+[Binomial(n-j-1, n-2*j)*Binomial(2*j, j): j in [0..Floor(n/2)]]): n in [0..40]]; // G. C. Greubel, Dec 12 2021

CoefficientList[Series[Sqrt[(1-x)/(1-x-4*x^2)], {x, 0, 40}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

[sum(binomial(n-j-1, n-2*j)*binomial(2*j, j) for j in (0..(n//2))) for n in [0..40]] # G. C. Greubel, Dec 12 2021

a(n) = A026584(n, n).
G.f.: sqrt((1-x)/(1-x-4*x^2)). - Ralf Stephan, Jan 08 2004
From Paul Barry, Jul 01 2009: (Start)
G.f.: 1/(1 -2*x^2/(1 -x -x^2/(1 -x^2/(1 -x -x^2/(1 -x^2/(1 -x -x^2/(1 - ... (continued fraction).
a(0) = 1, a(n) = Sum_{k=0..floor(n/2)} (k/(n-k))*C(n-k,k)*A000984(k). (End)
From Paul Barry, Mar 23 2011: (Start)
a(n) = Sum_{k=0..floor(n/2)} C(n-k-1,n-2*k)*A000984(k).
a(n) = Sum_{k=0..floor(n/2)} C(n-k-1,n-2*k)*C(2*k,k). (End)
D-finite with recurrence n*a(n) +2*(-n+1)*a(n-1) +(-3*n+2)*a(n-2) +2*(2*n-5)*a(n-3) = 0. - R. J. Mathar, Nov 24 2012
a(n) ~ (sqrt(17)+1)^(n-1/2) / (17^(1/4) * sqrt(Pi*n) * 2^(n-3/2)). - Vaclav Kotesovec, Feb 12 2014

A026599 a(n) = Sum_{j=0..2*i, i=0..n} A026584(i,j).

Original entry on oeis.org

1, 3, 9, 23, 61, 155, 401, 1023, 2629, 6723, 17241, 44135, 113101, 289643, 742049, 1900623, 4868821, 12471315, 31946601, 81831863, 209618269, 536945723, 1375418801, 3523201695, 9024876901, 23117683683, 59217191289, 151687926023

Offset: 0

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 0..1000
Amir Sapir, The Tower of Hanoi with Forbidden Moves, The Computer J. 47 (1) (2004) 20, case cyclic++, sequence c(n) (offset 1).
Index entries for linear recurrences with constant coefficients, signature (2,3,-4).

Cf. A026584, A026585, A026587, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026596, A026597 (first differences), A026598, A027282, A027283, A027284, A027285, A027286.

Cf. A006131, A026581.

[n le 3 select 3^(n-1) else 2*Self(n-1) +3*Self(n-2) -4*Self(n-3): n in [1..41]]; // G. C. Greubel, Dec 15 2021

LinearRecurrence[{2,3,-4}, {1,3,9}, 40] (* G. C. Greubel, Dec 15 2021 *)

[( (1+x)/((1-x)*(1-x-4*x^2)) ).series(x,n+1).list()[n] for n in (0..40)] # G. C. Greubel, Dec 15 2021

G.f.: (1+x)/((1-x)*(1-x-4*x^2)). - Ralf Stephan, Feb 04 2004
From Klaus Purath, Feb 02 2021: (Start)
a(n) = 2*a(n-1) + 3*a(n-2) - 4*a(n-3).
a(n) = Sum_{j=0..n} A026597(j). (End)
a(n) = 2^n*(Fibonacci(n+2, 1/2) + Fibonacci(n+1, 1/2)) - 1/2. - G. C. Greubel, Dec 15 2021

A026595 a(n) = T(n, floor(n/2)), where T is given by A026584.

Original entry on oeis.org

1, 1, 1, 1, 5, 8, 19, 22, 69, 121, 341, 406, 1203, 2155, 6336, 7624, 22593, 40717, 121483, 147001, 438533, 792351, 2381512, 2892044, 8677763, 15703156, 47419503, 57728737, 173984792, 315180458, 954961034, 1164727748, 3522101709

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, A026596, 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 *)
Table[T[n, Floor[n/2]], {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)
[T(n, n//2) for n in (0..40)] # G. C. Greubel, Dec 13 2021

a(n) = A026584(n, floor(n/2))

A026587 a(n) = T(n, n-2), T given by A026584. 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, 9, 28, 62, 167, 399, 1024, 2518, 6359, 15819, 39759, 99427, 249699, 626203, 1573524, 3953446, 9943905, 25019005, 62994733, 158680545, 399936573, 1008438757, 2543992514, 6420413940, 16210331727, 40943722115, 103453402718

Offset: 2

Clark Kimberling

nonn

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

Cf. A026584, A026585, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026596, 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 *)
Table[T[n, n-2], {n, 2, 40}] (* G. C. Greubel, Dec 12 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)
[T(n, n-2) for n in (2..40)] # G. C. Greubel, Dec 12 2021

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

Conjecture: (n+2)*a(n) = (3*n+2)*a(n-1) +(3*n+2)*a(n-2) -(11*n-16)*a(n-3) -2*(n-3)*a(n-4) +4*(2*n-9)*a(n-5). - R. J. Mathar, Jun 23 2013

A026589 a(n) = T(n,n-4), T given by A026584. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=4.

Original entry on oeis.org

1, 2, 9, 22, 69, 178, 497, 1294, 3452, 8964, 23430, 60556, 156663, 403214, 1037191, 2660978, 6821200, 17459732, 44657246, 114117628, 291449047, 743904326, 1897956899, 4840429962, 12340947855, 31455453822, 80158533099

Offset: 4

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 4..1000

Cf. A026584, A026585, A026587, A026590, A026591, A026592, A026593, A026594, A026595, A026596, 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 *)
Table[T[n, n-4], {n, 4, 40}] (* G. C. Greubel, Dec 12 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)
[T(n, n-4) for n in (4..40)] # G. C. Greubel, Dec 12 2021

a(n) = A026584(n, n-4).

Conjecture: -(n+4)*(65*n-269)*a(n) +(-65*n^2+140*n+1933)*a(n-1) +(809*n^2-2431*n-4514)*a(n-2) +(-123*n^2+2496*n-205)*a(n-3) +2*(-726*n^2+3737*n-4395)*a(n-4) +8*(56*n-215)*(2*n-9)*a(n-5) = 0. - R. J. Mathar, Jun 23 2013

A026590 a(n) = T(2*n, n), where T is given by A026584.

Original entry on oeis.org

1, 1, 5, 19, 69, 341, 1203, 6336, 22593, 121483, 438533, 2381512, 8677763, 47419503, 173984792, 954961034, 3522101709, 19397198595, 71831252031, 396646918211, 1473610012405, 8154682794333, 30376120747792, 168394714422722, 628648474795879, 3490216221862041, 13053833414221023, 72566287730964469

Offset: 0

Clark Kimberling

nonn

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

Cf. A026584, A026585, A026587, A026589, A026591, A026592, A026593, A026594, A026595, A026596, 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]= Block[{$RecursionLimit= Infinity}, T[2*n,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)
[T(2*n, n) for n in (0..40)] # G. C. Greubel, Dec 13 2021

a(n) = A026584(n, n).

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

A026592 a(n) = T(2*n, n-2), where T is given by A026584.

Original entry on oeis.org

1, 3, 14, 65, 251, 1288, 4830, 25518, 95388, 510532, 1910821, 10309234, 38656462, 209766714, 787912030, 4294635438, 16155375825, 88371236851, 332859949946, 1826080683788, 6885797551334, 37867515477338, 142929375411104, 787637258527505, 2975423924172735, 16425495119248041, 62096233990615140, 343318987947145114

Offset: 2

Clark Kimberling

nonn

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

Cf. A026584, A026585, A026587, A026589, A026590, A026591, A026593, A026594, A026595, A026596, 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]= Block[{$RecursionLimit= Infinity}, T[2*n,n-2]];
Table[a[n], {n, 2, 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)
[T(2*n, n-2) for n in (2..40)] # G. C. Greubel, Dec 13 2021

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

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

A026593 a(n) = T(2*n-1, n-1), where T is given by A026584.

Original entry on oeis.org

1, 1, 8, 22, 121, 406, 2155, 7624, 40717, 147001, 792351, 2892044, 15703156, 57728737, 315180458, 1164727748, 6385672193, 23691834033, 130316812494, 485018155062, 2674846358141, 9980763478121, 55161813337474, 206262229900060, 1142020843590221, 4277853480389546, 23721423518350124, 88991782850212510

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, A026594, A026595, A026596, 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]= Block[{$RecursionLimit= Infinity}, T[2*n-1,n-1]];
Table[a[n], {n, 1, 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)
[T(2*n-1, n-1) for n in (1..40)] # G. C. Greubel, Dec 13 2021

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

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

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

Clark Kimberling

nonn

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

Cf. A026584, A026585, A026587, A026589, A026590, A026591, A026592, A026593, A026595, A026596, 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*)
Table[T[2*n-1, n-2], {n, 2, 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)
[T(2*n-1, n-2) for n in (2..40)] # G. C. Greubel, Dec 13 2021

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

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

cp's OEIS Frontend

A026584 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; and for i >= 2 and j = 2..2i-2, T(i,j) = T(i-1,j-2) + T(i-1,j-1) + T(i-1,j) if i+j is odd, and T(i,j) = T(i-1,j-2) + T(i-1,j) if i+j is even.

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A026585 a(n) = T(n,n), T given by A026584. Also a(n) is the number of integer strings s(0), ..., s(n) counted by T, such that s(n)=0.

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A026599 a(n) = Sum_{j=0..2*i, i=0..n} A026584(i,j).

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A026595 a(n) = T(n, floor(n/2)), where T is given by A026584.

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A026587 a(n) = T(n, n-2), T given by A026584. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=2.

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Mathematica

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A026589 a(n) = T(n,n-4), T given by A026584. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=4.

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Crossrefs

Programs

Mathematica

Sage

Formula

A026590 a(n) = T(2*n, n), where T is given by A026584.

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

Sage

Formula

Extensions

A026592 a(n) = T(2*n, n-2), where T is given by A026584.

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