A026105 -id:A026105 - cp's OEIS frontend

A025566 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = sum of numbers in row n+1 of the array T defined in A026105. Also a(n) = T(n,n), where T is the array defined in A025564.

1, 1, 1, 3, 8, 22, 61, 171, 483, 1373, 3923, 11257, 32418, 93644, 271219, 787333, 2290200, 6673662, 19478091, 56930961, 166613280, 488176938, 1431878079, 4203938697, 12353600427, 36331804089, 106932444885, 314946659951, 928213563878

Offset: 0

Clark Kimberling

nonn

a(n+1) is the number of Motzkin (2n)-paths whose last weak valley occurs immediately after step n. A weak valley in a Motzkin path (A001006) is an interior vertex whose following step has nonnegative slope and whose preceding step has nonpositive slope. For example, the weak valleys in the Motzkin path F.UF.FD.UD occur after the first, third and fifth steps as indicated by the dots (U=upstep of slope 1, D=downstep of slope -1, F=flatstep of slope 0) and, with n=2, a(3)=3 counts FFUD, UDUD, UFFD. - David Callan, Jun 07 2006

Starting with offset 2: (1, 3, 8, 22, 61, 171, 483, ...), = row sums of triangle A136537. - Gary W. Adamson, Jan 04 2008

Michael De Vlieger, Table of n, a(n) for n = 0..2100
Jean-Luc Baril, Richard Genestier, Sergey Kirgizov, Pattern distributions in Dyck paths with a first return decomposition constrained by height, arXiv:1911.03119 [math.CO], 2019.
C. Dalfó, M. A. Fiol, and N. López, New results for the Mondrian art problem, arXiv:2007.09639 [math.CO], 2020.
D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012), #P33. - From _N. J. A. Sloane_, May 11 2012
Christian Krattenthaler, Daniel Yaqubi, Some determinants of path generating functions, II, arXiv:1802.05990 [math.CO], 2018; Adv. Appl. Math. 101 (2018), 232-265.
Donatella Merlini, Massimo Nocentini, Algebraic Generating Functions for Languages Avoiding Riordan Patterns, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.3.

First differences of A026135. Row sums of triangle A026105.
Pairwise sums of A005727. Column k=2 in A115990.
Cf. A001791, A132816.
Cf. A136537.

List([0..30],i->Sum([0..Int(i/2)],k->Binomial(i-2,k)*Binomial(i-k,k))); # Muniru A Asiru, Mar 09 2019

seq( sum('binomial(i-2,k)*binomial(i-k,k)', 'k'=0..floor(i/2)), i=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001

CoefficientList[Series[x+(2x(x-1))/(1-3x-Sqrt[1-2x-3x^2]),{x,0,30}],x] (* Harvey P. Dale, Jun 12 2016 *)

G.f.: x + 2*x*(x-1)/(1-3x-sqrt(1-2x-3x^2)); for n > 1, first differences of the "directed animals" sequence A005773: a(n) = A005773(n) - A005773(n-1). - Emeric Deutsch, Aug 16 2002
Starting (1, 3, 8, 22, 61, 171, ...) gives the inverse binomial transform of A001791 starting (1, 4, 15, 56, 210, 792, ...). - Gary W. Adamson, Sep 01 2007
a(n) is the sum of the (n-2)-th row of triangle A131816. - Gary W. Adamson, Sep 01 2007
D-finite with recurrence n*a(n) +(-3*n+2)*a(n-1) +(-n+2)*a(n-2) +3*(n-4)*a(n-3)=0. - R. J. Mathar, Sep 15 2020

A026109 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1, s(n) = 3, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = T(n,n-3), where T is the array defined in A026105.

Original entry on oeis.org

1, 3, 10, 30, 89, 259, 748, 2148, 6150, 17578, 50204, 143364, 409500, 1170300, 3346944, 9579840, 27444681, 78698475, 225887010, 648985414, 1866356437, 5372348487, 15478733108, 44637360700, 128837626255, 372183158061, 1076041247778

Offset: 3

Clark Kimberling

nonn

First differences of A005323. Cf. A026124.

G.f.: z(1-z)M^4, with M the g.f. of the Motzkin numbers (A001006).

Conjecture: (n+5)*a(n) +5*(-n-3)*a(n-1) +4*n*a(n-2) +8*n*a(n-3) +(-5*n+19)*a(n-4) +3*(-n+5)*a(n-5)=0. - R. J. Mathar, Jun 23 2013

A026110 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1, s(n) = 4, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = T(n,n-4), where T is the array defined in A026105.

Original entry on oeis.org

1, 4, 15, 50, 160, 496, 1509, 4530, 13475, 39820, 117117, 343278, 1003665, 2929200, 8537910, 24863724, 72363951, 210532540, 612398025, 1781252110, 5181318054, 15073505216, 43860668800, 127657036000, 371654416575, 1082359229796

Offset: 4

Clark Kimberling

nonn

Apparently the Motzkin transform of the 6th row of A025117, i.e., of 1, 4, 11, 20, ..., 11, 4, 1 followed by zeros. - R. J. Mathar, Dec 11 2008

First differences of A005324. Cf. A001006, A026125.

Cf. A025117

G.f.: z(1-z)M^5, with M the g.f. of the Motzkin numbers (A001006).

Conjecture: -(n+6)*(n-4)*a(n) +(4*n^2-n-51)*a(n-1) +(-2*n^2+11*n+18)*a(n-2) -(4*n-1)*(n-3)*a(n-3) +3*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Jun 23 2013

A026112 a(n) = T(2n,n), where T is the array defined in A026105.

Original entry on oeis.org

1, 1, 6, 30, 160, 873, 4851, 27300, 155115, 887910, 5112767, 29582886, 171857127, 1001762125, 5856231825, 34320863160, 201579530004, 1186231850298, 6992515244280, 41281814341780, 244049819182260, 1444558452300909, 8560106124666867, 50777065370076660

Offset: 0

Clark Kimberling

nonn

Bisection of A026117.

More terms from Alois P. Heinz, Sep 16 2019

A026117 T(n,[ n/2 ]), where T is the array defined in A026105.

Original entry on oeis.org

1, 1, 1, 2, 6, 10, 30, 50, 160, 265, 873, 1442, 4851, 7996, 27300, 44928, 155115, 254950, 887910, 1457852, 5112767, 8387170, 29582886, 48492028, 171857127, 281521541, 1001762125, 1640055250, 5856231825, 9582770760, 34320863160, 56135075250

Offset: 0

Clark Kimberling

nonn

A026111 a(n) = T(2n-1,n), where T is the array defined in A026105.

Original entry on oeis.org

3, 16, 89, 496, 2795, 15884, 90909, 523312, 3026850, 17577660, 102425257, 598575368, 3506930635, 20591787300, 121145018325, 713948286528, 4214030403902, 24907512574608, 147403116947718

Offset: 2

Clark Kimberling

nonn

A026113 T(3n,n), where T is the array defined in A026105.

Original entry on oeis.org

1, 2, 15, 112, 879, 7084, 58123, 482980, 4051542, 34236670, 291001515, 2485209984, 21308128100, 183304844334, 1581390987039, 13676468811804, 118533903449357, 1029280548493050, 8952719328878815, 77988115665259800, 680280714418808610, 5941248392350208706

Offset: 0

Clark Kimberling

nonn

Cf. A026105.

a(13) corrected and more terms from Sean A. Irvine, Sep 16 2019

A026114 T(4n,n), where T is the array defined in A026105.

Original entry on oeis.org

1, 3, 28, 275, 2834, 30020, 323817, 3537963, 39025125, 433641593, 4846850372, 54432520983, 613727606977, 6942918090500, 78767492060334, 895824168454107, 10210180642895226, 116591635975080084, 1333619552180129775, 15277391011739272885, 175247987032943618365

Offset: 0

Clark Kimberling

nonn

Cf. A026105.

More terms and offset corrected by Sean A. Irvine, Sep 16 2019

A026115 a(n) = T(2n,n-1), where T is the array defined in A026105.

Original entry on oeis.org

1, 3, 15, 77, 413, 2266, 12636, 71290, 405790, 2325923, 13406666, 77632590, 451270925, 2631763500, 15391191315, 90230338449, 530098757210, 3120158068370, 18395925762046, 108621483748778, 642235028537889, 3801900814767378, 22531392119093175, 133663824177577317

Offset: 1

Clark Kimberling

nonn

Cf. A026105.

More terms from Sean A. Irvine, Sep 16 2019

A026116 T(2n,n+1), where T is the array defined in A026105.

Original entry on oeis.org

1, 7, 44, 259, 1509, 8767, 50973, 296907, 1733150, 10138970, 59435705, 349085651, 2053897585, 12103791975, 71432963850, 422138418075, 2497699796394, 14794839261958, 87725376843892, 520654743222562, 3092804089136541, 18386738886807495, 109390940367285906, 651267699186002859

Offset: 1

Clark Kimberling

nonn

Cf. A026105.

More terms from Sean A. Irvine, Sep 16 2019

cp's OEIS Frontend

A025566 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = sum of numbers in row n+1 of the array T defined in A026105. Also a(n) = T(n,n), where T is the array defined in A025564.

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A026109 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1, s(n) = 3, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = T(n,n-3), where T is the array defined in A026105.

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A026110 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1, s(n) = 4, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = T(n,n-4), where T is the array defined in A026105.

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A026112 a(n) = T(2n,n), where T is the array defined in A026105.

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A026117 T(n,[ n/2 ]), where T is the array defined in A026105.

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A026111 a(n) = T(2n-1,n), where T is the array defined in A026105.

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A026113 T(3n,n), where T is the array defined in A026105.

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A026114 T(4n,n), where T is the array defined in A026105.

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A026115 a(n) = T(2n,n-1), where T is the array defined in A026105.

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A026116 T(2n,n+1), where T is the array defined in A026105.

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