A054444 -id:A054444 - cp's OEIS frontend

A268886 T(n,k)=Number of nXk binary arrays with some element plus some horizontally or antidiagonally adjacent neighbor totalling two exactly once.

0, 1, 0, 2, 5, 0, 5, 14, 20, 0, 10, 54, 84, 71, 0, 20, 158, 501, 462, 235, 0, 38, 475, 2190, 4133, 2418, 744, 0, 71, 1340, 9996, 27130, 31956, 12252, 2285, 0, 130, 3740, 42362, 186732, 317966, 236960, 60666, 6865, 0, 235, 10204, 178400, 1187838, 3283890

Offset: 1

R. H. Hardin, Feb 15 2016

Table starts
.0.....1.......2.........5..........10............20..............38
.0.....5......14........54.........158...........475............1340
.0....20......84.......501........2190..........9996...........42362
.0....71.....462......4133.......27130........186732.........1187838
.0...235....2418.....31956......317966.......3283890........31427480
.0...744...12252....236960.....3596174......55491832.......800733668
.0..2285...60666...1706732....39670270.....911930096.....19876401224
.0..6865..295230..12034000...429588382...14681855846....483987898760
.0.20284.1417452..83485488..4585939726..232688402028..11611969197776
.0.59155.6732102.571836176.48401059362.3642322709900.275345016177616

			Some solutions for n=4 k=4
..0..1..0..0. .0..0..0..0. .0..0..1..0. .0..0..0..0. .0..1..0..0
..0..0..0..1. .0..1..0..1. .1..0..0..1. .0..0..1..1. .0..1..0..0
..0..0..1..0. .1..0..0..0. .1..0..0..0. .0..0..0..0. .0..1..0..0
..1..0..1..0. .1..0..0..1. .1..1..0..1. .0..1..0..0. .1..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..799

Column 2 is A054444(n-1).

Row 1 is A001629.

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3) -a(n-4)
k=3: a(n) = 10*a(n-1) -31*a(n-2) +30*a(n-3) -9*a(n-4)
k=4: a(n) = 16*a(n-1) -88*a(n-2) +200*a(n-3) -208*a(n-4) +96*a(n-5) -16*a(n-6) for n>7
k=5: [order 8] for n>9
k=6: [order 10] for n>12
k=7: [order 14] for n>16
Empirical for row n:
n=1: a(n) = 2*a(n-1) +a(n-2) -2*a(n-3) -a(n-4)
n=2: a(n) = 2*a(n-1) +5*a(n-2) -4*a(n-3) -11*a(n-4) -6*a(n-5) -a(n-6)
n=3: [order 9]
n=4: [order 16]
n=5: [order 26]
n=6: [order 42]
n=7: [order 68]

A060920 Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zeros).

Original entry on oeis.org

1, 2, 1, 5, 5, 1, 13, 20, 9, 1, 34, 71, 51, 14, 1, 89, 235, 233, 105, 20, 1, 233, 744, 942, 594, 190, 27, 1, 610, 2285, 3522, 2860, 1295, 315, 35, 1, 1597, 6865, 12473, 12402, 7285, 2534, 490, 44, 1, 4181, 20284, 42447, 49963, 36122, 16407, 4578, 726, 54, 1

Offset: 0

Wolfdieter Lang, Apr 20 2001

Companion triangle (odd-indexed members) A060921.

			Triangle begins as:
      1;
      2,     1;
      5,     5,      1;
     13,    20,      9,      1;
     34,    71,     51,     14,      1;
     89,   235,    233,    105,     20,     1;
    233,   744,    942,    594,    190,    27,     1;
    610,  2285,   3522,   2860,   1295,   315,    35,    1;
   1597,  6865,  12473,  12402,   7285,  2534,   490,   44,    1;
   4181, 20284,  42447,  49963,  36122, 16407,  4578,  726,   54,  1;
  10946, 59155, 140109, 190570, 163730, 91959, 33705, 7776, 1035, 65, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Yidong Sun, Numerical Triangles and Several Classical Sequences, Fib. Quart. 43, no. 4, Nov. 2005, pp. 359-370.

Column sequences: A001519 (k=0), A054444 (k=1), A061178 (k=2), A061179 (k=3), A061180 (k=4), A061181 (k=5).

Cf. A000045, A001519, A007583, A037027, A060921, A061176.

A060920:= func< n,k | (&+[Binomial(2*n-k-j, j)*Binomial(2*n-k-2*j, k): j in [0..2*n-k]]) >;
[A060920(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 06 2021

A060920[n_, k_]:= Sum[Binomial[2*n-k-j, j]*Binomial[2*n-k-2*j, k], {j,0,2*n-k}];
Table[A060920[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 06 2021 *)

def A060920(n,k): return sum(binomial(2*n-k-j, j)*binomial(2*n-k-2*j, k) for j in (0..2*n-k))
flatten([[A060920(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 06 2021

T(n, k) = A037027(2*n-k, k).
T(n, k) = ((2*(n-k) + 1)*A060921(n-1, k-1) + 4*n*T(n-1, k-1))/(5*k), n >= k >= 1.
T(n, 0) = F(n)^2 + F(n+1)^2 = A001519(n), with the Fibonacci numbers F(n) = A000045(n).
Sum_{k=0..n} T(n, k) = (2^(2*n + 1) + 1)/3 = A007583(n).
G.f. for column m >= 0: x^m*pFe(m+1, x)/(1-3*x+x^2)^(m+1), where pFe(n, x) := Sum_{m=0..n} A061176(n, m)*x^m (row polynomials of signed triangle A061176).
G.f.: (1-x*(1+y))/(1 - (3+2*y)*x + (1+y)^2*x^2). - Vladeta Jovovic, Oct 11 2003

A027991 a(n) = Sum{T(n,k)*T(n,2n-k)}, 0<=k<=n-1, T given by A027926.

Original entry on oeis.org

1, 3, 12, 40, 130, 404, 1227, 3653, 10720, 31090, 89316, 254568, 720757, 2029095, 5684340, 15855964, 44061862, 122032508, 336966015, 927953705, 2549229256, 6987648358, 19115124552, 52194037200, 142274514025, 387215773899

Offset: 1

Clark Kimberling

nonn

From Wolfdieter Lang, Jan 02 2012: (Start)
a(n) = A024458(2*n-1), n>=1 (bisection, odd arguments).
chate(n):=a(n+1), n>=0, is the even part of the bisection of the half-convolution of the sequence A000045(n+1), n>=0, with itself. See a comment on A201204 for the definition of half-convolution. There one finds also the rule for the o.g.f.s of the bisection. Here the o.g.f. of the sequence chate(n), n>=0, is Chate(x):= (Ce(x)+U2(x))/2 with Ce(x)=(1-x+x^2)/(1-3*x+x^2)^2, the o.g.f. of A054444(n), and
  U2(x)=(1-x)/((1+x)*(1-3*x+x^2)), the o.g.f. of A007598(n+1), n>=0. This results (after multiplying with x) in the o.g.f. given below in the formula section. It is equivalent to the explicit formula given there, as can be seen after a partial fraction decomposition of the o.g.f.
(End)

Cf. A027926, A024458, A201204, A007598.

a(n) = (1/5)[n*F(2n+2) - n*F(2n-2) + F(2n-1) - (-1)^n], F(n)=A000045(n).

O.g.f.: x*(1-2*x+2*x^2)/((1-3*x+x^2)^2*(1+x)). See the comment above. - Wolfdieter Lang, Jan 02 2012

A061178 Third column (m=2) of triangle A060920 (bisection of Fibonacci triangle, even part).

Original entry on oeis.org

1, 9, 51, 233, 942, 3522, 12473, 42447, 140109, 451441, 1426380, 4434420, 13599505, 41225349, 123723351, 368080793, 1086665562, 3186317718, 9286256393, 26916587307, 77634928209, 222920650081

Offset: 0

Wolfdieter Lang, Apr 20 2001

Numerator polynomial of g.f. is sum(A061176(3,m)*x^m,m=0..3).

A001519, A054444.

a(n)= A060920(n+2, 2).

G.f.: (1-x^3)/(1-3*x+x^2)^3.

A121530 Number of double rises at an odd level in all nondecreasing Dyck paths of semilength n. A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing.

Original entry on oeis.org

0, 1, 4, 14, 47, 148, 454, 1359, 4004, 11644, 33521, 95696, 271300, 764605, 2143964, 5985186, 16643779, 46124692, 127433562, 351106955, 964976460, 2646158176, 7241414949, 19779499584, 53933402472, 146828245753, 399137621524

Offset: 1

Emeric Deutsch, Aug 05 2006

nonn

a(n)=Sum(k*A121529(n,k), k>=0). a(n)+A121532(n)=A054444(n-2).

			a(3)=4 because we have UDUDUD, UDU/UDD, U/UDDUD, U/UDUDD and U/UUDDD, the double rises at an odd level being indicated by a / (U=(1,1), D=(1,-1)).

E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
Index entries for linear recurrences with constant coefficients, signature (6,-9,-5,15,-1,-4,1)

Cf. A121529, A121532, A054444.

g:=z^2*(1-2*z-z^2+4*z^3-3*z^4)/(1+z)/(1-3*z+z^2)^2/(1-z-z^2): gser:=series(g,z=0,33): seq(coeff(gser,z,n),n=1..30);

Rest[CoefficientList[Series[x^2*(1-2*x-x^2+4*x^3-3*x^4)/(1+x)/(1-3*x+x^2)^2 /(1-x-x^2), {x, 0, 20}], x]] (* Vaclav Kotesovec, Mar 20 2014 *)

G.f.=z^2*(1-2z-z^2+4z^3-3z^4)/[(1+z)(1-3z+z^2)^2*(1-z-z^2)].
a(n) ~ (3-sqrt(5)) * (3+sqrt(5))^n * n / (5 * 2^(n+1)). - Vaclav Kotesovec, Mar 20 2014
Equivalently, a(n) ~ phi^(2*n-2) * n / 5, where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 06 2021

A121532 Number of double rises at an even level in all nondecreasing Dyck paths of semilength n. A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing.

Original entry on oeis.org

0, 0, 1, 6, 24, 87, 290, 926, 2861, 8640, 25634, 75015, 217100, 622620, 1772097, 5011394, 14093980, 39448623, 109954398, 305344314, 845165725, 2332485420, 6420202246, 17629525871, 48304680504, 132092031672, 360557665825

Offset: 1

Emeric Deutsch, Aug 05 2006

nonn

			a(3)=1 because we have UDUDUD, UDUUDD, UUDDUD, UUDUDD and UU/UDDD, the double rises at an odd level being indicated by a / (U=(1,1), D=(1,-1)).

E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
Index entries for linear recurrences with constant coefficients, signature (6,-9,-5,15,-1,-4,1).

Cf. A121530, A121531, A054444.

R:=PowerSeriesRing(Integers(), 30); [0,0] cat Coefficients(R!( x^3*(1-3*x^2+2*x^3-x^4)/((1+x)*(1-3*x+x^2)^2*(1-x-x^2)) )); // G. C. Greubel, May 24 2019

g:=z^3*(1-3*z^2+2*z^3-z^4)/(1+z)/(1-3*z+z^2)^2/(1-z-z^2): gser:=series(g,z=0,35): seq(coeff(gser,z,n),n=1..32);

Rest[CoefficientList[Series[x^3*(1-3*x^2+2*x^3-x^4)/(1+x)/(1-3*x+x^2)^2/(1-x-x^2), {x, 0, 30}], x]] (* Vaclav Kotesovec, Mar 20 2014 *)

my(x='x+O('x^30)); concat([0,0], Vec(x^3*(1-3*x^2+2*x^3-x^4)/((1+x)*(1-3*x+x^2)^2*(1-x-x^2)))) \\ G. C. Greubel, May 24 2019

a=(x^3*(1-3*x^2+2*x^3-x^4)/((1+x)*(1-3*x+x^2)^2*(1-x-x^2)) ).series(x, 30).coefficients(x, sparse=False); a[1:] # G. C. Greubel, May 24 2019

a(n) = Sum_{k>=0} k*A121531(n,k).
a(n) = A054444(n-2) - A121530(n).
G.f.: x^3*(1-3*x^2+2*x^3-x^4)/((1+x)*(1-3*x+x^2)^2*(1-x-x^2)). [Corrected by Georg Fischer, May 24 2019]
a(n) ~ (3-sqrt(5)) * (3+sqrt(5))^n * n / (5 * 2^(n+1)). - Vaclav Kotesovec, Mar 20 2014
Equivalently, a(n) ~ phi^(2*n-2) * n / 5, where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 06 2021

cp's OEIS Frontend

A268886 T(n,k)=Number of nXk binary arrays with some element plus some horizontally or antidiagonally adjacent neighbor totalling two exactly once.

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A060920 Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zeros).

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A027991 a(n) = Sum{T(n,k)*T(n,2n-k)}, 0<=k<=n-1, T given by A027926.

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A061178 Third column (m=2) of triangle A060920 (bisection of Fibonacci triangle, even part).

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A121530 Number of double rises at an odd level in all nondecreasing Dyck paths of semilength n. A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing.

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A121532 Number of double rises at an even level in all nondecreasing Dyck paths of semilength n. A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing.

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