A003169 -id:A003169 - cp's OEIS frontend

A365766 Expansion of (1/x) * Series_Reversion( x*(1-x)^5/(1+x) ).

1, 6, 56, 626, 7721, 101322, 1387648, 19606874, 283711805, 4183074796, 62618441024, 949174260118, 14539621490403, 224721722650224, 3500129695446816, 54882906729334378, 865664769346769005, 13725517938819785298, 218639429113140366968

Offset: 0

Seiichi Manyama, Sep 18 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..811

Cf. A003169, A006318, A365764, A365765.

Cf. A365753.

a(n) = sum(k=0, n, binomial(n+1, k)*binomial(6*n-k+4, n-k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+1,k) * binomial(6*n-k+4,n-k) = (1/(n+1)) * Sum_{k=0..n} binomial(5*n+k+4,k) * binomial(n+1,n-k).

A211788 Triangle enumerating certain two-line arrays of positive integers.

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 1, 7, 21, 21, 1, 10, 47, 126, 126, 1, 13, 82, 324, 818, 818, 1, 16, 126, 642, 2300, 5594, 5594, 1, 19, 179, 1107, 4977, 16741, 39693, 39693, 1, 22, 241, 1746, 9335, 38642, 124383, 289510, 289510, 1, 25, 312, 2586, 15941, 77273, 301630, 939880, 2157150, 2157150

Offset: 1

Peter Bala, Aug 02 2012

This is the table of f(n,k) in the notation of Carlitz (p.123). The triangle enumerates two-line arrays of positive integers
............a_1 a_2 ... a_n..........
............b_1 b_2 ... b_n..........
such that
1) max(a_i, b_i) <= min(a_(i+1), b_(i+1)) for 1 <= i <= n-1
2) max(a_i, b_i) <= i for 1 <= i <= n
3) a_n = b_n = k.
See A071948 and A193091 for other two-line array enumerations.
It appears that the row reverse array is the Riordan array (f(x), g(x)), where f(x) = 1 + x + 4*x^2 + 21*x^3 + 126*x^4 + 818*x^5 + ... is the g.f. of A003168 and g(x) = x + 3*x^2 + 14*x^3 + 79*x^4 + 494*x^5 + 3294*x^6 + ... is the g.f. of A003169. - Peter Bala, Nov 26 2024

			Triangle begins
.n\k.|..1....2....3....4....5....6
= = = = = = = = = = = = = = = = = =
..1..|..1
..2..|..1....1
..3..|..1....4....4
..4..|..1....7...21...21
..5..|..1...10...47..126..126
..6..|..1...13...82..324..818..818
...
T(4,2) = 7: The 7 two-line arrays are
...1 1 1 2....1 1 2 2....1 2 2 2....1 1 1 2
...1 1 1 2....1 1 2 2....1 2 2 2....1 1 2 2
...........................................
...1 1 2 2....1 1 2 2....1 2 2 2...........
...1 1 1 2....1 2 2 2....1 1 2 2...........

L. Carlitz, Enumeration of two-line arrays, Fib. Quart., Vol. 11 Number 2 (1973), 113-130.

Cf. A003168 (main diagonal), A211789 (row sums).

Cf. A003169, A071948, A193091.

Recurrence equation:
T(1,1) = 1; T(n,n) = T(n,n-1); T(n+1,k) = Sum_{j = 1..k} (2*k-2*j+1)*T(n,j) for 1 <= k <= n.
T(n+1,k+1) = (1/n) * ((n - k)*Sum_{i = 0..k} C(n, k-i)*C(2*n+i, i) + Sum_{i = 1..k} C(n, k-i)*C(2*n+i, i-1)).
Row reverse has production matrix
  1 1
  3 3 1
  5 5 3 1
  7 7 5 3 1
  ...
Main diagonal T(n,n) = A003168(n). Row sums A211789.

A365765 Expansion of (1/x) * Series_Reversion( x*(1-x)^4/(1+x) ).

Original entry on oeis.org

1, 5, 39, 365, 3772, 41491, 476410, 5644477, 68493324, 846937140, 10633195119, 135185288475, 1736883987836, 22516798984946, 294169295918996, 3869084306851933, 51189853304834940, 680816769653570044, 9097058255214149068, 122064057533865334100

Offset: 0

Seiichi Manyama, Sep 18 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..865

Cf. A003169, A006318, A365764, A365766.

Cf. A365752.

CoefficientList[(1/x) *InverseSeries[Series[x*(1-x)^4/(1+x),{x,0,20}]],x] (* Stefano Spezia, May 04 2025 *)

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

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

A379174 G.f. A(x) satisfies A(x) = (1 + xA(x)^3)/(1 - xA(x))^2.

Original entry on oeis.org

1, 3, 20, 176, 1772, 19309, 221651, 2640016, 32322122, 404256442, 5142846467, 66341063274, 865723122919, 11408144684248, 151593390664710, 2029025599194394, 27330120599494110, 370183683091079836, 5038997387800717228, 68896081533831380702, 945747379824209853435

Offset: 0

Seiichi Manyama, Dec 17 2024

nonn

Cf. A003169, A249924, A379173.

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

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

A100328 Column 1 of triangle A100326, in which row n equals the inverse binomial of column n of square array A100324, with leading zero omitted.

Original entry on oeis.org

1, 4, 20, 116, 736, 4952, 34716, 250868, 1855520, 13979192, 106901032, 827644424, 6474611984, 51100656544, 406400018092, 3253636464756, 26201323746880, 212093247874904, 1724793778005528, 14084738953391768, 115447965121881856

Offset: 0

Paul D. Hanna, Nov 17 2004

Self-convolution of A100327, which equals the row sums of triangle A100326.

Vaclav Kotesovec, Table of n, a(n) for n = 0..650

Cf. A003168, A003169, A100324, A100326, A100327.

{a(n)=if(n==0,1,sum(j=0,n, if(j==0,1,sum(k=0,j,2*binomial(j,k)*binomial(2*j+k,k-1)/j))* if(n-j==0,1,sum(k=0,n-j,2*binomial(n-j,k)*binomial(2*n-2*j+k,k-1)/(n-j)))))}

G.f.: (1+G003169(x))*G003169(x)/x, where G003169(x) is the g.f. of A003169.
Recurrence: 4*(n+1)*(2*n+1)*(17*n^2 - 28*n + 12)*a(n) = (1207*n^4 - 1988*n^3 + 1013*n^2 - 124*n - 12)*a(n-1) - 2*(n-2)*(2*n-3)*(17*n^2 + 6*n + 1)*a(n-2). - Vaclav Kotesovec, Jul 05 2014
a(n) ~ sqrt(95+393/sqrt(17)) * ((71+17*sqrt(17))/16)^n / (4*sqrt(2*Pi) * n^(3/2)). - Vaclav Kotesovec, Jul 05 2014
From Peter Bala, Sep 08 2024: (Start)
a(n) = (2/n) * Sum_{k = 0..n} binomial(n+1, n-k-1)*binomial(2*n, k)*2^(n-k) for n >= 1.
a(n) = 4*Jacobi_P(n-1, 2, n+1, 3)/n for n >= 1. Cf. A003168. (End)

A379173 G.f. A(x) satisfies A(x) = (1 + x)/(1 - x*A(x))^2.

Original entry on oeis.org

1, 3, 11, 53, 284, 1630, 9794, 60830, 387390, 2515892, 16599051, 110943779, 749603067, 5111606801, 35133394554, 243146923574, 1692918638012, 11850006727400, 83341778073920, 588646472669454, 4173607638548291, 29694593381322531, 211941668053441490, 1517087043428034420

Offset: 0

Seiichi Manyama, Dec 17 2024

nonn

Cf. A003169, A249924, A379174.

Cf. A025227, A379171.

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

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

A294724 Number of Q graphs with 2*n vertices rooted at an internal edge.

Original entry on oeis.org

0, 0, 0, 1, 11, 96, 785, 6283, 49987, 397768, 3174084, 25426290, 204538114, 1652327820, 13402442651, 109129475455, 891793354235, 7312118010384, 60141701374424, 496095022537998, 4103183675407098, 34022136992384720

Offset: 1

Sean A. Irvine, Nov 07 2017

nonn

R. C. Read, On the enumeration of a class of plane multigraphs, Aequat. Math. 31 (1986) no 1, 47-63.

Cf. A003168, A003169.

G.f.: P(x) * (B(x) - x)^2 where P(x) is the g.f. for A003169 and B(x) is the g.f. for A003168.

A007164 Number of P-graphs with 2n edges.

Original entry on oeis.org

1, 2, 7, 31, 167, 999, 6495, 44619, 319463, 2356406, 17775821, 136405506, 1060866006, 8339982193, 66147792753, 528559162227, 4250413757583, 34368723106860, 279254390054538, 2278795583070154, 18667432144042678, 153452484745198722

Offset: 1

N. J. A. Sloane

nonn

R. C. Read, On the enumeration of a class of plane multigraphs, Aequat. Math., 31 (1986), 47-63.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Cf. A003169, A007163, A007165.

a(n) = (A003169(n) + 2 * A007165(n) + A007163(n)) / 4. - Sean A. Irvine, Nov 06 2017

a(5), a(9), a(10) corrected and more terms from Sean A. Irvine, Nov 06 2017

A007171 Number of Q-graphs with 2n edges.

Original entry on oeis.org

1, 2, 4, 16, 89, 579, 3989, 28630, 210865, 1584308, 12091902, 93483120, 730503054, 5760438853, 45780146521, 366304396662, 2948412461685, 23857049238752, 193944769559906, 1583294537879106, 12974430494985262, 106685110047676402

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. C. Read, On the enumeration of a class of plane multigraphs, Aequat. Math. 31 (1986) no 1, 47-63.

Cf. A003169, A007165, A007170.

a(1) = 1, a(n) = A007170(n) + (A003169(n-1) + A007165(n-1)) / 2. - Sean A. Irvine, Nov 07 2017

a(9) corrected and more terms from Sean A. Irvine, Nov 07 2017

cp's OEIS Frontend

A365766 Expansion of (1/x) * Series_Reversion( x*(1-x)^5/(1+x) ).

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A211788 Triangle enumerating certain two-line arrays of positive integers.

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A365765 Expansion of (1/x) * Series_Reversion( x*(1-x)^4/(1+x) ).

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A379174 G.f. A(x) satisfies A(x) = (1 + x*A(x)^3)/(1 - x*A(x))^2.

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A100328 Column 1 of triangle A100326, in which row n equals the inverse binomial of column n of square array A100324, with leading zero omitted.

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A379173 G.f. A(x) satisfies A(x) = (1 + x)/(1 - x*A(x))^2.

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A294724 Number of Q graphs with 2*n vertices rooted at an internal edge.

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A007164 Number of P-graphs with 2n edges.

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A007171 Number of Q-graphs with 2n edges.

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A379174 G.f. A(x) satisfies A(x) = (1 + xA(x)^3)/(1 - xA(x))^2.