A071969 -id:A071969 - cp's OEIS frontend

A119255 Duplicate of A071969.

1, 1, 2, 6, 19, 63, 219, 787, 2897, 10869, 41414, 159822, 623391, 2453727, 9733866, 38877318, 156206233, 630947421, 2560537092, 10435207116, 42689715279, 175243923783, 721649457417, 2980276087005, 12340456995177, 51222441676513

Offset: 0

dead

A071946 Triangle T(n,k) read by rows giving number of underdiagonal lattice paths from (0,0) to (n,k) using only steps R = (1,0), V = (0,1) and D = (3,1).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 4, 6, 6, 1, 6, 13, 19, 19, 1, 8, 23, 44, 63, 63, 1, 10, 37, 87, 156, 219, 219, 1, 12, 55, 155, 330, 568, 787, 787, 1, 14, 77, 255, 629, 1260, 2110, 2897, 2897, 1, 16, 103, 395, 1111, 2527, 4856, 7972, 10869, 10869, 1, 18, 133, 583, 1849, 4706, 10130, 18889, 30545, 41414, 41414

Offset: 0

N. J. A. Sloane, Jun 15 2002

			Triangle T(n,k) begins:
  1;
  1, 1;
  1, 2,  2;
  1, 4,  6,  6;
  1, 6, 13, 19, 19;
  ...

Alois P. Heinz, Rows n = 0..150, flattened
D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad J. Math., 49 (1997), 301-320.

Related arrays: A071943, A071944, A071945.

A108076 is the reverse, A119254 is the row sums and A071969 is the last (largest) number in each row.

T:= proc(n, k) option remember; `if`(n=0 and k=0, 1,
     `if`(k<0 or nAlois P. Heinz, May 05 2023

T[n_, k_] := T[n, k] = If[n == 0 && k == 0, 1,
   If[k < 0 || n < k, 0, T[n-1, k] + T[n, k-1] + T[n-3, k-1]]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 25 2025, after Alois P. Heinz *)

More terms from Joshua Zucker, May 10 2006

A369265 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1+x^3) ).

Original entry on oeis.org

1, 2, 7, 31, 153, 806, 4439, 25250, 147193, 874732, 5279635, 32276245, 199439761, 1243633652, 7815804351, 49455190791, 314807497953, 2014530780524, 12952334769203, 83628832755779, 542022781854953, 3525150296312984, 22998642171764363, 150478455899387966

Offset: 0

Seiichi Manyama, Jan 18 2024

nonn

Cf. A369267, A369269.

Cf. A071969, A370247.

A369265 := proc(n)
    add(binomial(n+1,k) * binomial(3*n-3*k+1,n-3*k),k=0..floor(n/3)) ;
    %/(n+1) ;
end proc;
seq(A369265(n),n=0..70) ; # R. J. Mathar, Jan 25 2024

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^2/(1+x^3))/x)

a(n, s=3, t=1, u=2) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(n+1,k) * binomial(3*n-3*k+1,n-3*k).
D-finite with recurrence 16*(n+1)*(2*n+1)*a(n) +4*(-89*n^2+15*n+2)*a(n-1) +3*(345*n^2-603*n+274)*a(n-2) +18*(-41*n^2+45*n+94)*a(n-3) +54*(-4*n^2+57*n-137)*a(n-4) +486*(n-4)*(n-5)*a(n-5) -243*(n-4)*(n-5)*a(n-6)=0. - R. J. Mathar, Jan 25 2024
a(n) = (1/(n+1)) * [x^n] ( 1/(1-x)^2 * (1+x^3) )^(n+1). - Seiichi Manyama, Feb 14 2024

A071944 Triangle read by rows giving numbers of paths in a lattice satisfying certain conditions.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 5, 6, 1, 4, 9, 16, 19, 1, 5, 14, 31, 54, 63, 1, 6, 20, 52, 111, 188, 219, 1, 7, 27, 80, 197, 405, 676, 787, 1, 8, 35, 116, 320, 752, 1508, 2492, 2897, 1, 9, 44, 161, 489, 1276, 2900, 5712, 9361, 10869, 1, 10, 54, 216, 714, 2034, 5095, 11296, 21933, 35702, 41414

Offset: 0

N. J. A. Sloane, Jun 15 2002

			Triangle begins with:
  1;
  1,   1;
  1,   2,   2;
  1,   3,   5,   6;
  1,   4,   9,  16,  19;
  1,   5,  14,  31,  54,  63;
  1,   6,  20,  52, 111, 188, 219;
  1,   7,  27,  80, 197, 405, 676, 787;
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened
D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad J. Math., 49 (1997), 301-320.

Diagonal entries form A071969.

[[((n-k+1)/(n+1))*(&+[Binomial(n+1, j)*Binomial(n+k -3*j, n): j in [0..Floor(k/3)]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Mar 17 2019

a := proc(n,k) if k<=n then (n-k+1)*sum(binomial(n+1,i)*binomial(n+k-3*i,n),i=0..k/3)/(n+1) else 0 fi end;

Table[((n-k+1)/(n+1))*Sum[Binomial[n+1, j]*Binomial[n+k-3*j, n], {j, 0, k/3}], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 17 2019 *)

{T(n,k) = ((n-k+1)/(n+1))*sum(j=0, floor(k/3), binomial(n+1, j)* binomial(n+k -3*j, n))}; \\ G. C. Greubel, Mar 17 2019

[[((n-k+1)/(n+1))*sum(binomial(n+1,j)*binomial(n+k-3*j,n) for j in (0..floor(k/3))) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 17 2019

T(n, k) = ((n-k+1)/(n+1))*Sum_{i=0..k/3} binomial(n+1, i)*binomial(n+k -3*i, n), for k <= n.

More terms from Emeric Deutsch, Dec 19 2003

A369268 Expansion of (1/x) * Series_Reversion( x * (1-x) / (1+x^3)^3 ).

Original entry on oeis.org

1, 1, 2, 8, 29, 105, 414, 1695, 7046, 29853, 128644, 561262, 2474142, 11006108, 49343508, 222715440, 1011217425, 4615519083, 21165513228, 97467424198, 450541090701, 2089777230606, 9723511785608, 45371996501895, 212271904284993, 995513843930049, 4679212044797252

Offset: 0

Seiichi Manyama, Jan 18 2024

nonn

Cf. A071969, A369266.

Cf. A369269, A369270.

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)/(1+x^3)^3)/x)

a(n, s=3, t=3, u=1) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(3*n+3,k) * binomial(2*n-3*k,n-3*k).

a(n) = (1/(n+1)) * [x^n] ( 1/(1-x) * (1+x^3)^3 )^(n+1). - Seiichi Manyama, Feb 14 2024

A119254 Row sums of A071946.

Original entry on oeis.org

1, 2, 5, 17, 58, 202, 729, 2695, 10140, 38719, 149682, 584672, 2304045, 9149194, 36573273, 147057039, 594374148, 2413480053, 9840832968, 40276235226, 165403090815, 681373222191, 2814872996190, 11659083772986, 48407568680323, 201431186725778, 839913707978789

Offset: 0

Joshua Zucker, May 10 2006

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A071946 is the triangle and A071969 has the last number of each row.

A369266 Expansion of (1/x) * Series_Reversion( x * (1-x) / (1+x^3)^2 ).

Original entry on oeis.org

1, 1, 2, 7, 24, 84, 313, 1209, 4769, 19166, 78253, 323570, 1352122, 5701467, 24229122, 103663575, 446163435, 1930390329, 8391341664, 36630504952, 160509484616, 705750073063, 3112865367660, 13769327908980, 61066953746400, 271488240652950, 1209671359828154

Offset: 0

Seiichi Manyama, Jan 18 2024

nonn

Cf. A071969, A369268.

Cf. A369267, A370248.

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)/(1+x^3)^2)/x)

a(n, s=3, t=2, u=1) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(2*n+2,k) * binomial(2*n-3*k,n-3*k).

a(n) = (1/(n+1)) * [x^n] ( 1/(1-x) * (1+x^3)^2 )^(n+1). - Seiichi Manyama, Feb 14 2024

A073187 Triangle of C(n+1,k)C(2n-3k,n-3k)/(n+1) by rows.

Original entry on oeis.org

1, 1, 2, 5, 1, 14, 5, 42, 21, 132, 84, 3, 429, 330, 28, 1430, 1287, 180, 4862, 5005, 990, 12, 16796, 19448, 5005, 165, 58786, 75582, 24024, 1430, 208012, 293930, 111384, 10010, 55, 742900, 1144066, 503880, 61880, 1001, 2674440, 4457400, 2238390

Offset: 1

Michael Somos, Jul 19 2002

			1;
1;
2;
5,  1;
14, 5;
42, 21;
132, 84, 3;
429, 330, 28;
1430, 1287, 180; ...

G. C. Greubel, Rows n=0..150 of triangle, flattened

Row sums give A071969.

T[n_, k_]:= If[k < 0 || k > n/3, 0, Binomial[n + 1, k]*Binomial[2*n - 3*k, n - 3*k]/(n + 1)]; Table[T[n, k], {n,0,10}, {k,0,Floor[n/3]}]//Flatten (* G. C. Greubel, May 29 2018 *)

alias(C,binomial); T(n,k)=if(k<0 || k>n/3,0,C(n+1,k)*C(2*n-3*k,n-3*k)/(n+1))

A365267 G.f. satisfies A(x) = 1 + xA(x)^2(1 + x^3*A(x)).

Original entry on oeis.org

1, 1, 2, 5, 15, 47, 153, 513, 1763, 6177, 21981, 79224, 288611, 1061019, 3931320, 14666135, 55041855, 207668702, 787225265, 2996851140, 11452198368, 43915195973, 168930713580, 651708006690, 2520840672423, 9774511167507, 37985839339052

Offset: 0

Seiichi Manyama, Aug 30 2023

nonn

Cf. A001002, A071969.

Cf. A063021, A365268.

a(n) = sum(k=0, n\4, binomial(n-3*k, k)*binomial(2*n-5*k+1, n-3*k)/(2*n-5*k+1));

a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k,k) * binomial(2*n-5*k+1,n-3*k)/(2*n-5*k+1).

A366025 Expansion of (1/x) * Series_Reversion( x*(1-x)/(1+x^5) ).

Original entry on oeis.org

1, 1, 2, 5, 14, 43, 139, 465, 1595, 5577, 19804, 71228, 258946, 950030, 3513050, 13079920, 48993149, 184490361, 698020080, 2652192675, 10115878915, 38717526745, 148655862210, 572412768275, 2209969761924, 8553073927858, 33176952295730, 128960722306128

Offset: 0

Seiichi Manyama, Sep 26 2023

nonn

Cf. A006318, A052709, A071969, A365268.

Cf. A365760, A366024.

CoefficientList[InverseSeries[Series[x(1-x)/(1+x^5),{x,0,28}],x]/x,x] (* Stefano Spezia, Sep 26 2023 *)

a(n) = sum(k=0, n\5, binomial(n-4*k, k)*binomial(2*n-5*k+1, n-4*k)/(2*n-5*k+1));

Vec(serreverse(x*(1-x)/(1+x^5)+O(x^30))/x) \\ Michel Marcus, Sep 26 2023

G.f. satisfies A(x) = 1 + x*A(x)^2*(1 + x^4*A(x)^3).

a(n) = Sum_{k=0..floor(n/5)} binomial(n-4*k,k) * binomial(2*n-5*k+1,n-4*k)/(2*n-5*k+1) = (1/(n+1)) * Sum_{k=0..floor(n/5)} binomial(n+1,k) * binomial(2*n-5*k,n-5*k).

cp's OEIS Frontend

A119255 Duplicate of A071969.

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A071946 Triangle T(n,k) read by rows giving number of underdiagonal lattice paths from (0,0) to (n,k) using only steps R = (1,0), V = (0,1) and D = (3,1).

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A369265 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1+x^3) ).

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A071944 Triangle read by rows giving numbers of paths in a lattice satisfying certain conditions.

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A369268 Expansion of (1/x) * Series_Reversion( x * (1-x) / (1+x^3)^3 ).

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A119254 Row sums of A071946.

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A369266 Expansion of (1/x) * Series_Reversion( x * (1-x) / (1+x^3)^2 ).

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A073187 Triangle of C(n+1,k)*C(2*n-3*k,n-3*k)/(n+1) by rows.

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A365267 G.f. satisfies A(x) = 1 + x*A(x)^2*(1 + x^3*A(x)).

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A073187 Triangle of C(n+1,k)C(2n-3k,n-3k)/(n+1) by rows.

A365267 G.f. satisfies A(x) = 1 + xA(x)^2(1 + x^3*A(x)).