A007317 -id:A007317 - cp's OEIS frontend

A378327 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(nk,k) / ((n-1)k + 1).

1, 2, 5, 25, 257, 4361, 104425, 3241316, 123865313, 5628753361, 296671566941, 17798975341467, 1197924420178381, 89394126594968755, 7326377073291002147, 654215578855903951141, 63225054646397348577601, 6575059243843086616460321, 732138834180570978286488133

Offset: 0

Vaclav Kotesovec, Nov 23 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..338

Cf. A007317, A007556, A188687, A346646, A346647, A346648, A346649, A346650.

Cf. A378326.

Table[Sum[Binomial[n, k] Binomial[n*k, k]/((n-1)*k + 1), {k, 0, n}], {n, 0, 20}]

a(n) ~ exp(n + exp(-1) - 1/2) * n^(n - 5/2) / sqrt(2*Pi).

A038392 Number of mono-4-polyhexes with n cells.

Original entry on oeis.org

1, 1, 2, 6, 19, 71, 274, 1117, 4650, 19819, 85710, 375712, 1664203, 7439593, 33515758, 152019560, 693625265, 3181528275, 14661581030, 67850297506, 315187646601, 1469195636293, 6869889703638, 32215399021901, 151467334017864, 713881817440421, 3372142139764434

Offset: 1

N. J. A. Sloane

J. Brunvoll, B. N. Cyvin, and S. J. Cyvin, Studies of some chemically relevant polygonal systems: mono-q-polyhexes, ACH Models in Chem., 133 (3) (1996), 277-298; see Eq. 16.

Robert Israel, Table of n, a(n) for n = 1..1437
S. J. Cyvin, Graph-theoretical studies on fluoranthenoids and fluorenoids. Part 1, Journal of Molecular Structure (Theochem), 262 (1992), 219-231.
N. Cyvin, E. Brendsdal, J. Brunvoll, S. J. Cyvin, A class of polygonal systems representing polycyclic conjugated hydrocarbons: Catacondensed monoheptafusenes, Monat. f. Chemie, 125 (1994), 1327-1337.
S. J. Cyvin, B. N. Cyvin, J. Brunvoll and E. Brendsdal, Enumeration and Classification of Certain Polygonal Systems Representing Polycyclic Conjugated Hydrocarbons: Annelated Catafusenes, Journal of Chemical Information and Modeling [formerly, J. Chem. Inform. Comput. Sci.], 34 (1994), pp. 1174-1180.
S. J. Cyvin, B. N. Cyvin, J. Brunvoll, Zhang Fuji, Guo Xiaofeng, and R. Tosic, Graph-theoretical studies on fluoranthenoids and fluorenoids: enumeration of some catacondensed systems, Journal of Molecular Structures (Theochem), 285 (1993), 179-185.
F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.
Eric Weisstein's World of Mathematics, Fusene.

Apart from initial term, (A002212 + A007317)/2. See A044045 for another version.

f:= gfun:-rectoproc({(250*n^2-250*n)*a(n)+(-300*n^2-150*n)*a(n+1)+(-325*n^2-875*n-600)*a(n+2)+(475*n^2+2045*n+2100)*a(n+3)+(35*n^2+265*n+540)*a(n+4)+(-193*n^2-1691*n-3660)*a(n+5)+(49*n^2+563*n+1596)*a(n+6)+(17*n^2+211*n+648)*a(n+7)+(-9*n^2-135*n-504)*a(n+8)+(n^2+17*n+72)*a(n+9), a(0) = 0, a(1) = 1, a(2) = 1, a(3) = 2, a(4) = 6, a(5) = 19, a(6) = 71, a(7) = 274, a(8) = 1117},a(n),remember):
map(f, [$1..50]); # Robert Israel, Oct 08 2017

f[z_] := Sqrt[5*z^2 - 6*z + 1]; g[z_] := (2*(1 - z^2) - (1-z)*f[z] - f[z^2])/ (4*(1-z)); Drop[ CoefficientList[ Series[ g[z], {z, 0, 24}], z], 1] (* Jean-François Alcover, Oct 13 2011, after Emeric Deutsch *)

G.f.: (2(1-z^2) - (1-z)f(z) - f(z^2))/(4(1-z)) where f(z) = sqrt(1-6z+5z^2). - Emeric Deutsch, Mar 14 2004

(250*n^2-250*n)*a(n)+(-300*n^2-150*n)*a(n+1)+(-325*n^2-875*n-600)*a(n+2)+(475*n^2+2045*n+2100)*a(n+3)+(35*n^2+265*n+540)*a(n+4)+(-193*n^2-1691*n-3660)*a(n+5)+(49*n^2+563*n+1596)*a(n+6)+(17*n^2+211*n+648)*a(n+7)+(-9*n^2-135*n-504)*a(n+8)+(n^2+17*n+72)*a(n+9) = 0. - Robert Israel, Oct 08 2017

More terms from Emeric Deutsch, Mar 14 2004

A086620 Symmetric square table of coefficients, read by antidiagonals, where T(n,k) is the coefficient of x^ny^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xyf(x,y)^2.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 14, 7, 1, 1, 9, 28, 28, 9, 1, 1, 11, 47, 79, 47, 11, 1, 1, 13, 71, 175, 175, 71, 13, 1, 1, 15, 100, 331, 504, 331, 100, 15, 1, 1, 17, 134, 562, 1196, 1196, 562, 134, 17, 1, 1, 19, 173, 883, 2464, 3514, 2464, 883, 173, 19, 1, 1, 21, 217

Offset: 0

Paul D. Hanna, Jul 24 2003

Determinants of upper left n X n matrices results in A086619: {1,2,10,150,7650,1438200,1051324200,...}, which is the products of the first n terms of the binomial transform of Catalan numbers (A007317): {1,2,5,15,51,188,731,2950,...}.

			Rows begin:
1,_1,__1,__1,___1,____1,____1,_____1, ...
1,_3,__5,__7,___9,___11,___13,____15, ...
1,_5,_14,_28,__47,___71,__100,___134, ...
1,_7,_28,_79,_175,__331,__562,___883, ...
1,_9,_47,175,_504,_1196,_2464,__4572, ...
1,11,_71,331,1196,_3514,_8764,_19244, ...
1,13,100,562,2464,_8764,26172,_67740, ...
1,15,134,883,4572,19244,67740,204831, ...

Cf. A086621 (diagonal), A086622 (antidiagonal sums), A086619 (determinants).

Contribution from Paul Barry, Feb 04 2009: (Start)
T(n,k)=sum{j=0..n+k, C(k,j-k)*C(n+2k-j,k)*if(k<=j,A000108(n-k),0)};
Regarded as a number triangle read by row, columns are generated by sum{j=0..k, C(k,j)*A000108(j)*x^j}*x^k/(1-x)^(k+1). (End)

A091698 Matrix inverse of triangle A063967.

Original entry on oeis.org

1, -1, 1, 1, -3, 1, -1, 8, -5, 1, 1, -23, 19, -7, 1, -1, 74, -69, 34, -9, 1, 1, -262, 256, -147, 53, -11, 1, -1, 993, -986, 615, -265, 76, -13, 1, 1, -3943, 3935, -2571, 1235, -431, 103, -15, 1, -1, 16178, -16169, 10862, -5591, 2216, -653, 134, -17, 1, 1

Offset: 0

Christian G. Bower, Jan 29 2004

Riordan array (1/(1+x), (sqrt(1+6x+5x^2)-x-1)/(2(1+x))). The absolute value array is (1/(1-x),xc(x)/(1-xc(x))) where c(x) is the g.f. of A000108. It factorizes as (1/(1-x),x/(1-x))(1,xc(x)). - Paul Barry, Jun 10 2005

			From _Paul Barry_, Apr 15 2010: (Start)
Triangle begins
  1,
  -1, 1,
  1, -3, 1,
  -1, 8, -5, 1,
  1, -23, 19, -7, 1,
  -1, 74, -69, 34, -9, 1,
  1, -262, 256, -147, 53, -11, 1,
  -1, 993, -986, 615, -265, 76, -13, 1,
  1, -3943, 3935, -2571, 1235, -431, 103, -15, 1
Production matrix begins
  -1, 1,
  0, -2, 1,
  0, 1, -2, 1,
  0, -1, 1, -2, 1,
  0, 1, -1, 1, -2, 1,
  0, -1, 1, -1, 1, -2, 1,
  0, 1, -1, 1, -1, 1, -2, 1,
  0, -1, 1, -1, 1, -1, 1, -2, 1,
  0, 1, -1, 1, -1, 1, -1, 1, -2, 1,
  0, -1, 1, -1, 1, -1, 1, -1, 1, -2, 1 (End)

Lara K. Pudwell, Ascent sequences and the binomial convolution of Catalan numbers, arXiv preprint arXiv:1408.6823 [math.CO], 2014.

Row sums: A091699. Row sums (absolute values): A007317. Column 1: A050511.

rows = 11; t[n_, k_] := Sum[Binomial[j, n - j]*Binomial[j, k], {j, 0, n}]; T = Table[t[n, k], {n, 0, rows - 1}, {k, 0, rows - 1}] // Inverse; Table[ T[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 11 2017 *)

A106534 Sum array of Catalan numbers (A000108) read by upward antidiagonals.

Original entry on oeis.org

1, 2, 1, 5, 3, 2, 15, 10, 7, 5, 51, 36, 26, 19, 14, 188, 137, 101, 75, 56, 42, 731, 543, 406, 305, 230, 174, 132, 2950, 2219, 1676, 1270, 965, 735, 561, 429, 12235, 9285, 7066, 5390, 4120, 3155, 2420, 1859, 1430, 51822, 39587, 30302, 23236, 17846, 13726, 10571, 8151, 6292, 4862

Offset: 0

Philippe Deléham, May 30 2005

The underlying array A is A(n, k) = Sum_{j=0..n} binomial(n, j)*A000108(k+j), n >= 0, k>= 0. See the example section. - Wolfdieter Lang, Oct 04 2019

			From _Wolfdieter Lang_, Oct 04 2019: (Start)
The triangle T(n, k) begins:
n\k      0      1      2      3     4     5     6     7     8     9    10 ...
0:       1
1:       2      1
2:       5      3      2
3:      15     10      7      5
4:      51     36     26     19    14
5:     188    137    101     75    56    42
6:     731    543    406    305   230   174   132
7:    2950   2219   1676   1270   965   735   561   429
8:   12235   9285   7066   5390  4120  3155  2420  1859  1430
9:   51822  39587  30302  23236 17846 13726 10571  8151  6292  4862
10: 223191 171369 131782 101480 78244 60398 46672 36101 27950 21658 16796
... reformatted and extended.
-------------------------------------------------------------------------
The array A(n, k) begins:
n\k  0   1    2    3     4     5      6 ...
-------------------------------------------
0:   1   1    2    5    14    42    132 ... A000108
1    2   3    7   19    56   174    561 ... A005807
2:   5  10   26   75   230   735   2420 ...
3:  15  36  101  305   965  3155  10571 ...
4:  51 137  406 1270  4120 13726  46672 ...
5: 188 543 1676 5390 17846 60398 207963 ...
... (End)

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Paul Barry and A. Hennessy, The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences, J. Int. Seq. 13 (2010) # 10.8.2, page 5.

Columns: A007317, A002212, see also A045868, A055452-A055455.
Diagonals: A000108, A005807.
Cf. A059346 (Catalan difference array as triangle).

function T(n,k)
  if k gt n then return 0;
  elif k eq n then return Catalan(n);
  else return T(n-1, k) + T(n, k+1);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 18 2021

# Uses floating point, precision might have to be adjusted.
C := n -> binomial(2*n,n)/(n+1);
H := (n,k) -> hypergeom([k-n,k+1/2],[k+2],-4);
T := (n,k) -> C(k)*H(n,k);
seq(print(seq(round(evalf(T(n,k),32)),k=0..n)),n=0..7);
# Peter Luschny, Aug 16 2012

T[n_, n_] := CatalanNumber[n]; T[n_, k_] /; 0 <= k < n := T[n-1, k] + T[n, k+1]; T[, ] = 0; Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Jean-François Alcover, Jun 11 2019 *)

def T(n, k) :
    if k > n : return 0
    if n == k : return binomial(2*n, n)/(n+1)
    return T(n-1, k) + T(n, k+1)
A106534 = lambda n,k: T(n, k)
for n in (0..5): [A106534(n,k) for k in (0..n)] # Peter Luschny, Aug 16 2012

T(n, k) = 0 if k > n; T(n, n) = A000108(n); T(n, k) = T(n-1, k) + T(n, k+1) if 0 <= k < n.
T(n, k) = binomial(2*k,k)/(k+1)*hypergeometric([k-n, k+1/2], [k+2], -4). - Peter Luschny, Aug 16 2012
T(n, k) = A(n-k, k) = Sum_{j=0..n-k} binomial(n-k, j)*A000108(k+j), n >= 0, k = 0..n. - Wolfdieter Lang, Oct 03 2019
G.f.: (sqrt(1-4*x*y)-sqrt((5*x-1)/(x-1)))/(2*x*(x*y-y+1)). - Vladimir Kruchinin, Jan 12 2024

A300865 Signed recurrence over binary enriched p-trees: a(n) = (-1)^(n-1) + Sum_{x + y = n, 0 < x <= y < n} a(x) * a(y).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 2, 4, 6, 10, 16, 27, 46, 77, 131, 224, 391, 672, 1180, 2050, 3626, 6344, 11276, 19863, 35479, 62828, 112685, 200462, 360627, 644199, 1162296, 2083572, 3768866, 6777314, 12289160, 22158106, 40255496, 72765144, 132453122, 239936528, 437445448

Offset: 1

Gus Wiseman, Mar 13 2018

nonn

Cf. A000992, A001190, A007317, A063834, A099323, A196545, A220418, A273866, A273873, A289501, A290261, A300352, A300442, A300443, A300862, A300863, A300864, A300866.

a[n_]:=a[n]=(-1)^(n-1)+Sum[a[k]*a[n-k],{k,1,n/2}];
Array[a,50]

A346074 a(n) = 1 + Sum_{k=0..n-5} a(k) * a(n-k-5).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 9, 14, 21, 30, 41, 59, 89, 136, 205, 301, 443, 664, 1011, 1545, 2341, 3530, 5341, 8143, 12487, 19148, 29299, 44817, 68721, 105742, 163025, 251392, 387595, 597988, 924047, 1430167, 2215595, 3433788, 5323915, 8260652, 12829849

Offset: 0

Ilya Gutkovskiy, Jul 04 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A007317, A090344, A216604, A307972, A346073.

a[n_] := a[n] = 1 + Sum[a[k] a[n - k - 5], {k, 0, n - 5}]; Table[a[n], {n, 0, 44}]
nmax = 44; A[] = 0; Do[A[x] = 1/(1 - x) + x^5 A[x]^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

a(n) = sum(k=0, n\5, binomial(n-4*k, k)*binomial(2*k, k)/(k+1)); \\ Seiichi Manyama, Jan 22 2023

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x^5 * A(x)^2.
Conjecture D-finite with recurrence (n+5)*a(n) +2*(-n-4)*a(n-1) +(n+3)*a(n-2) +2*(-2*n+5)*a(n-5) +4*(n-3)*a(n-6)=0. - R. J. Mathar, Feb 17 2022
a(n) = Sum_{k=0..floor(n/5)} binomial(n-4*k,k) * Catalan(k). - Seiichi Manyama, Jan 22 2023

A351972 a(n) = 1 + Sum_{k=0..floor((n-1)/2)} a(k) * a(n-2*k-1).

Original entry on oeis.org

1, 2, 3, 6, 11, 21, 40, 78, 151, 294, 572, 1115, 2172, 4234, 8252, 16088, 31361, 61140, 119191, 232370, 453010, 883167, 1721768, 3356675, 6543988, 12757830, 24871992, 48489172, 94531974, 184294706, 359291464, 700456240, 1365573493, 2662252082, 5190190005

Offset: 0

Ilya Gutkovskiy, Feb 26 2022

nonn

Cf. A000621, A007317, A351973.

a[n_] := a[n] = 1 + Sum[a[k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 34}]
nmax = 34; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[x^2])) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

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

A360102 a(n) = Sum_{k=0..n} binomial(n+2k,n-k) Catalan(k).

Original entry on oeis.org

1, 2, 7, 30, 141, 703, 3655, 19603, 107679, 602756, 3426049, 19721069, 114728723, 673494466, 3984493735, 23732956453, 142204128507, 856560123504, 5183708936061, 31502904805922, 192180259402691, 1176416604202925, 7223943302003917, 44486888142708088

Offset: 0

Seiichi Manyama, Jan 25 2023

nonn

Partial sums of A360100.
Partial sums are A258973.
Cf. A000108, A162481.
Cf. A006318, A007317, A162476, A360103.

A360102 := proc(n)
    add(binomial(n+2*k,n-k)*A000108(k),k=0..n) ;
end proc:
seq(A360102(n),n=0..70) ; # R. J. Mathar, Mar 12 2023

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

my(N=30, x='x+O('x^N)); Vec(2/((1-x)*(1+sqrt(1-4*x/(1-x)^3))))

G.f. A(x) satisfies A(x) = 1/(1-x) + x * A(x)^2 / (1-x)^2.
G.f.: (1/(1-x)) * c(x/(1-x)^3), where c(x) is the g.f. of A000108.
D-finite with recurrence (n+1)*a(n) +4*(-2*n+1)*a(n-1) +10*(n-2)*a(n-2) +2*(-2*n+7)*a(n-3) +(n-5)*a(n-4)=0. - R. J. Mathar, Mar 12 2023

A360103 a(n) = Sum_{k=0..n} binomial(n+4k,n-k) Catalan(k).

Original entry on oeis.org

1, 2, 9, 49, 283, 1715, 10793, 69906, 463031, 3122264, 21363065, 147951489, 1035173405, 7306326465, 51959150713, 371950057003, 2678083379707, 19381867703946, 140915907625531, 1028760981192771, 7538511404971231, 55427326349613665, 408789584900354397

Offset: 0

Seiichi Manyama, Jan 25 2023

nonn

Partial sums of A360101.
Cf. A000108, A360057.
Cf. A006318, A007317, A162476, A360102.

A360103 := proc(n)
    add(binomial(n+4*k,n-k)*A000108(k),k=0..n) ;
end proc:
seq(A360103(n),n=0..40) ; # R. J. Mathar, Mar 12 2023

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

my(N=30, x='x+O('x^N)); Vec(2/((1-x)*(1+sqrt(1-4*x/(1-x)^5))))

G.f. A(x) satisfies A(x) = 1/(1-x) + x * A(x)^2 / (1-x)^4.
G.f.: (1/(1-x)) * c(x/(1-x)^5), where c(x) is the g.f. of A000108.
D-finite with recurrence (n+1)*a(n) +2*(-5*n+3)*a(n-1) +(19*n-47)*a(n-2) +20*(-n+4)*a(n-3) +5*(3*n-17)*a(n-4) +2*(-3*n+22)*a(n-5) +(n-9)*a(n-6)=0. - R. J. Mathar, Mar 12 2023

cp's OEIS Frontend

A378327 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n*k,k) / ((n-1)*k + 1).

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A038392 Number of mono-4-polyhexes with n cells.

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Maple

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Formula

Extensions

A086620 Symmetric square table of coefficients, read by antidiagonals, where T(n,k) is the coefficient of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xy*f(x,y)^2.

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A091698 Matrix inverse of triangle A063967.

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A106534 Sum array of Catalan numbers (A000108) read by upward antidiagonals.

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Magma

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Mathematica

Sage

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A300865 Signed recurrence over binary enriched p-trees: a(n) = (-1)^(n-1) + Sum_{x + y = n, 0 < x <= y < n} a(x) * a(y).

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A346074 a(n) = 1 + Sum_{k=0..n-5} a(k) * a(n-k-5).

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PARI

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A351972 a(n) = 1 + Sum_{k=0..floor((n-1)/2)} a(k) * a(n-2*k-1).

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Mathematica

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A360102 a(n) = Sum_{k=0..n} binomial(n+2*k,n-k) * Catalan(k).

A378327 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(nk,k) / ((n-1)k + 1).

A086620 Symmetric square table of coefficients, read by antidiagonals, where T(n,k) is the coefficient of x^ny^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xyf(x,y)^2.

A360102 a(n) = Sum_{k=0..n} binomial(n+2k,n-k) Catalan(k).

A360103 a(n) = Sum_{k=0..n} binomial(n+4k,n-k) Catalan(k).