A162148 -id:A162148 - cp's OEIS frontend

A175136 Triangle T(n,k) read by rows: number of LCO forests of size n with k leaves, 1 <= k <= n.

1, 1, 1, 2, 2, 1, 4, 6, 3, 1, 8, 17, 12, 4, 1, 16, 46, 44, 20, 5, 1, 32, 120, 150, 90, 30, 6, 1, 64, 304, 482, 370, 160, 42, 7, 1, 128, 752, 1476, 1412, 770, 259, 56, 8, 1, 256, 1824, 4344, 5068, 3402, 1428, 392, 72, 9, 1, 512, 4352, 12368, 17285, 14000, 7168, 2436

Offset: 1

R. J. Mathar, Feb 21 2010

From Johannes W. Meijer, May 06 2011: (Start)
The Row1, Kn11, Kn12, Kn13, Kn21, Kn22, Kn23, Kn3, Kn4 and Ca1 triangle sums link A175136 with several sequences, see the crossrefs. For the definitions of these triangle sums see A180662.
It is remarkable that the coefficients of the right hand columns of A175136, and subsequently those of triangle A175136, can be generated with the aid of the row coefficients of A091894. For the fourth, fifth and sixth right hand columns see A162148, A190048 and A190049. The a(n) formulas of the right hand columns lead to an explicit formula for the T(n,k), see the formulas and the second Maple program. (End)
Triangle T(n,k), 1 <= k <= n, read by rows, given by (0,1,1,0,1,1,0,1,1,0,1,1,0,1,...) DELTA (1,0,0,1,0,0,1,0,0,1,0,0,1,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 29 2011.
T(n,k) is the number of noncrossing partitions of n containing k runs, where a block forms a run if it consists of an interval of integers. For example, T(4,2)=6 counts 1/234, 12/34, 123/4, 1/24/3, 13/2/4, 14/2/3. - David Callan, Oct 14 2012

			Triangle starts
    1;
    1,    1;
    2,    2,    1;
    4,    6,    3,    1;
    8,   17,   12,    4,    1;
   16,   46,   44,   20,    5,    1;
   32,  120,  150,   90,   30,    6,    1;
   64,  304,  482,  370,  160,   42,    7,    1;
  128,  752, 1476, 1412,  770,  259,   56,    8,    1;
Triangle (0,1,1,0,1,1,0,...) DELTA (1,0,0,1,0,0,1,...) begins:
  1;
  0,  1;
  0,  1,  1;
  0,  2,  2,  1;
  0,  4,  6,  3,  1;
  0,  8, 17, 12,  4,  1; ... - _Philippe Deléham_, Oct 29 2011

David Callan, A bijection on Dyck paths and its cycle structure, El. J. Combinat. 14 (2007) # R28.
David Callan, On Ascent, Repetition and Descent Sequences, arXiv:1911.02209 [math.CO], 2019.
David Callan and Emeric Deutsch, The Run Transform, arXiv:1112.3639 [math.CO], 2011.
K. Manes, A. Sapounakis, I. Tasoulas, and P. Tsikouras, Nonleft peaks in Dyck paths: a combinatorial approach, Discrete Math., 337 (2014), 97-105.

Triangle columns: A000012, A000027, A002378, A162148, A190048, A190049, A011782, A190050, A190051.
Triangle sums (see the comments): A000108 (Row1), A005043 (Related to Kn11, Kn12, Kn13 and Kn4), A007477 (Related to Kn21, Kn22, Kn23 and Kn3), A099251 (Kn4), A166300 (Ca1). - Johannes W. Meijer, May 06 2011
Cf. A000108 (row sums), A196182

lco := proc(siz,leav) (1-(1-4*x*(1-x)/(1-x*y))^(1/2))/2/x ; coeftayl(%,x=0,siz ) ; coeftayl(%,y=0,leav ) ; end proc: seq(seq(lco(n,k),k=1..n),n=1..9) ;
T := proc(n, k): add(A091894(n-k, k1)*binomial(n-k1-1, n-k), k1=0..floor((n-k)/2)) end: A091894 := proc(n, k): if n=0 and k=0 then 1 elif n=0 then 0 else 2^(n-2*k-1)* binomial(n-1, 2*k) * binomial(2*k, k)/(k+1) fi end: seq(seq(T(n, k), k=1..n), n=1..10); # Johannes W. Meijer, May 06 2011, revised Nov 23 2012

A091894[n_, k_] := 2^(n - 2*k - 1)*Binomial[n - 1, 2*k]*(Binomial[2*k, k]/(k + 1)); t[n_, k_] := Sum[A091894[n - k, k1]*Binomial [n - k1 - 1, n - k], {k1, 0, (n - k)/2}]; t[n_, n_] = 1; Table[t[n, k], {n, 1, 11}, {k, 1, n}] // Flatten(* Jean-François Alcover, Jun 13 2013, after Johannes W. Meijer *)

G.f.: (1-(1-4*x*(1-x)/(1-x*y))^(1/2))/(2*x).

T(n,k) = Sum_{k1=0..floor((n-k)/2)} A091894(n-k, k1)*binomial(n-k1-1, n-k), 1 <= k <= n. - Johannes W. Meijer, May 06 2011

Variable names changed by Johannes W. Meijer, May 06 2011

A254407 a(n) = n(n+1)(11*n +10)/6.

Original entry on oeis.org

0, 7, 32, 86, 180, 325, 532, 812, 1176, 1635, 2200, 2882, 3692, 4641, 5740, 7000, 8432, 10047, 11856, 13870, 16100, 18557, 21252, 24196, 27400, 30875, 34632, 38682, 43036, 47705, 52700, 58032, 63712, 69751, 76160, 82950, 90132, 97717, 105716, 114140, 123000

Offset: 0

Bruno Berselli, Jan 30 2015

Similar sequences of the type m*P(s,m) - Sum_{i=1..m} P(s-1,i), where P(s,m) is the m-th s-gonal number:
s=3: A027480(n) = (n+1)*A000217(n+1) - Sum_{i=1..n+1} i;
s=4: A162148(n) = (n+1)*A000290(n+1) - Sum_{i=1..n+1} A000217(i);
s=5: A245301(n) = (n+1)*A000326(n+1) - Sum_{i=1..n+1} A000290(i);
s=6: A085788(n) = (n+1)*A000384(n+1) - Sum_{i=1..n+1} A000326(i);
s=7:       a(n) = (n+1)*A000566(n+1) - Sum_{i=1..n+1} A000384(i).

			532 is the 7th term because A000566(7)=112 and Sum_{i=1..7} A000384(i)=252, therefore 7*112-252 = 532.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wikipedia, Polygonal numbers: Table of values.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A011875, A027480, A057145, A085788, A132112, A162148, A245301, A254963.

Magma
```
[n*(n+1)*(11*n+10)/6: n in [0..40]];
```

A254407:= n-> n*(n+1)*(11*n+10)/6; seq(A254407(n), n=0..50); # G. C. Greubel, Mar 31 2021

Table[n (n + 1) (11 n + 10)/6, {n, 0, 40}]
Column[CoefficientList[Series[x (7 + 4 x) / (1 - x)^4, {x, 0, 60}], x]] (* Vincenzo Librandi, Jan 31 2015 *)

makelist(n*(n+1)*(11*n+10)/6, n, 0, 40);

vector(40, n, n--; n*(n+1)*(11*n+10)/6)

Sage
```
[n*(n+1)*(11*n+10)/6 for n in (0..40)]
```

G.f.: x*(7 + 4*x)/(1 - x)^4.
a(-n) = -A132112(n-1).
a(n) = Sum_{k=0..n} A011875(11*k+2).
Equivalently, partial sums of A254963.
E.g.f.: x*(42 + 54*x + 11*x^2)*exp(x)/6. - G. C. Greubel, Mar 31 2021

A177254 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} having k adjacent blocks (0 <= k <= n). An adjacent block is a block of the form (i, i+1, i+2, ...).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 1, 4, 6, 3, 1, 5, 13, 17, 12, 4, 1, 21, 51, 61, 44, 20, 5, 1, 91, 219, 255, 185, 90, 30, 6, 1, 422, 1019, 1182, 867, 440, 160, 42, 7, 1, 2103, 5108, 5964, 4430, 2322, 896, 259, 56, 8, 1, 11226, 27448, 32373, 24406, 13118, 5292, 1638, 392, 72, 9, 1

Offset: 0

Emeric Deutsch, May 07 2010

Sum of entries in row n = A000110(n) (the Bell numbers).

			T(4,2)=6 because we have 1-234, 12-34, 123-4, 13-2-4, 14-2-3, and 1-24-3.
Triangle starts:
     1;
     0,    1;
     0,    1,    1;
     0,    2,    2,    1;
     1,    4,    6,    3,    1;
     5,   13,   17,   12,    4,   1;
    21,   51,   61,   44,   20,   5,   1;
    91,  219,  255,  185,   90,  30,   6,  1;
   422, 1019, 1182,  867,  440, 160,  42,  7,  1;
  2103, 5108, 5964, 4430, 2322, 896, 259, 56,  8,  1;

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A000110, A002378, A162148, A168444, A177255, A177256, A177257.

Cf. A302560.

Q[0] := 1: for n to 10 do Q[n] := expand(u*subs(w = v, diff(Q[n-1], u))+u*subs(w = v, diff(Q[n-1], v))+w*(diff(Q[n-1], w))+w*subs(w = v, Q[n-1])) end do: for n from 0 to 10 do P[n] := sort(expand(subs({v = t, w = t, u = 1}, Q[n]))) end do; for n from 0 to 10 do seq(coeff(P[n], t, j), j = 0 .. n) end do; # yields sequence in triangular form

The row generating polynomial P[n](t) is given by P[n](t)=Q[n](1,t,t), where Q[n](u,v,w) is obtained recursively from Q[n](u,v,w) =u(dQ[n-1]/du){w=v} + u(dQ[n-1]/dv){w=v} + w(dQ[n-1]/dw) + w(Q[n-1])_{w=v}, Q[0]=1. Here Q[n](u,v,w) is the trivariate generating polynomial of the partitions of {1,2,...,n}, where u marks blocks that are not adjacent, v marks adjacent blocks not ending with n, and w marks adjacent blocks ending with n.
T(n, 0) = A168444(n).
Sum_{k=0..n} T(n, k) = A000110(n) (row sums).
Sum_{k=0..n} k*T(n, k) = A177255(n).
From G. C. Greubel, May 12 2024: (Start)
T(n, n) = 1.
T(n, n-1) = n-1, for n >= 1.
T(n, n-2) = A002378(n-2), for n >= 2.
T(n, n-3) = A162148(n-3), for n >= 3.
T(n, n-4) = A302560(n-3), for n >= 4. (End)

A190048 Expansion of (8+6*x)/(1-x)^5.

Original entry on oeis.org

8, 46, 150, 370, 770, 1428, 2436, 3900, 5940, 8690, 12298, 16926, 22750, 29960, 38760, 49368, 62016, 76950, 94430, 114730, 138138, 164956, 195500, 230100, 269100, 312858, 361746, 416150, 476470, 543120, 616528, 697136, 785400, 881790, 986790, 1100898

Offset: 0

Johannes W. Meijer, May 06 2011

Equals the fifth right hand column of A175136.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A175136, A162148, A190049.

Related to A000332 and A091894.

[(7*n^4+58*n^3+173*n^2+218*n+96)/12: n in [0..50]]; // Vincenzo Librandi, May 07 2011

A190048 := proc(n) option remember; a(n):=(7*n^4+58*n^3+173*n^2+218*n+96)/12 end: seq(A190048(n),n=0..35);

LinearRecurrence[{5,-10,10,-5,1}, {8,46,150,370,770}, 30] (* or *) CoefficientList[Series[(8+6*x)/(1-x)^5, {x, 0, 50}], x] (* G. C. Greubel, Jan 10 2018 *)

x='x+O('x^30); Vec((8+6*x)/(1-x)^5) \\ G. C. Greubel, Jan 10 2018

for(n=0,50, print1((7*n^4 +58*n^3 +173*n^2 +218*n +96)/12, ", ")) \\ G. C. Greubel, Jan 10 2018

G.f.: (8+6*x)/(1-x)^5.
a(n) = 8*binomial(n+4,4) + 6*binomial(n+3,4).
a(n) = A091894(4,0)*binomial(n+4,4) + A091894(4,1)*binomial(n+3,4).
a(n) = (7*n^4 +58*n^3 +173*n^2 +218*n +96)/12.

A190049 Expansion of (16+24x+2x^2)/(x-1)^6.

Original entry on oeis.org

16, 120, 482, 1412, 3402, 7168, 13692, 24264, 40524, 64504, 98670, 145964, 209846, 294336, 404056, 544272, 720936, 940728, 1211098, 1540308, 1937474, 2412608, 2976660, 3641560, 4420260, 5326776, 6376230, 7584892

Offset: 0

Johannes W. Meijer, May 06 2011

Equals the sixth right hand column of A175136.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A175136, A162148, A190048.

Related to A000389 and A091894.

[(21*n^5+245*n^4+1105*n^3+2395*n^2+2474*n+960)/60: n in [0..50]]; // Vincenzo Librandi, May 07 2011

A190049 := proc(n) option remember; a(n):=(21*n^5 +245*n^4 +1105*n^3 +2395*n^2 +2474*n +960)/60 end: seq(A190049(n),n=0..27);

LinearRecurrence[{6,-15,20,-15,6,-1}, {16,120,482,1412,3402,7168}, 30] (* or *) CoefficientList[Series[(16 +24*x +2*x^2)/(1-x)^6, {x, 0, 50}], x] (* G. C. Greubel, Jan 10 2018 *)

x='x+O('x^30); Vec((16 +24*x +2*x^2)/(1-x)^6) \\ G. C. Greubel, Jan 10 2018

for(n=0,30, print1((21*n^5 +245*n^4 +1105*n^3 +2395*n^2 +2474*n +960)/60, ", ")) \\ G. C. Greubel, Jan 10 2018

G.f.: (16 +24*x +2*x^2)/(1-x)^6.
a(n) = 16*binomial(n+5,5) +24*binomial(n+4,5) +2*binomial(n+3,5).
a(n) = A091894(5,0)*binomial(n+5,5) + A091894(5,1)*binomial(n+4,5) + A091894(5,2)*binomial(n+3,5).
a(n) = (21*n^5 +245*n^4 +1105*n^3 +2395*n^2 +2474*n +960)/60.

A126264 a(n) = 5n^2 + 3n.

Original entry on oeis.org

8, 26, 54, 92, 140, 198, 266, 344, 432, 530, 638, 756, 884, 1022, 1170, 1328, 1496, 1674, 1862, 2060, 2268, 2486, 2714, 2952, 3200, 3458, 3726, 4004, 4292, 4590, 4898, 5216, 5544, 5882, 6230, 6588, 6956, 7334, 7722, 8120, 8528, 8946, 9374, 9812, 10260

Offset: 1

Gary W. Adamson, Dec 22 2006

			a(24) = 5*24^2 + 3*24 = 2880 + 72 = 2952.

L. B. W. Jolley, Summation of Series, Dover Publications, 1961, p. 12

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001622, A147875.

a:=n->5*n^2+3*n: seq(a(n),n=1..55); # Emeric Deutsch, Apr 17 2007

Table[n*(5*n + 3), {n,1,50}] (* G. C. Greubel, Aug 23 2017 *)

a(n)=5*n^2+3*n \\ Charles R Greathouse IV, Jun 17 2017

Sum_{i=1..n} a(i) = n*(n+1)*(5n+7)/3 = 2*A162148(n).
a(n) = 2*A147875(n+1).
From G. C. Greubel, Aug 23 2017: (Start)
G.f.: 2*x*(x + 4)/(1 - x)^3.
E.g.f.: x*(5*x + 8)*exp(x). (End)
Sum_{n>=1} 1/a(n) = 5/9 + sqrt(1-2/sqrt(5))*Pi/6 + log(phi)*sqrt(5)/6 - 5*log(5)/12, where phi is the golden ratio (A001622). - Amiram Eldar, Aug 21 2022

More terms from Emeric Deutsch, Apr 17 2007

cp's OEIS Frontend

A175136 Triangle T(n,k) read by rows: number of LCO forests of size n with k leaves, 1 <= k <= n.

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A254407 a(n) = n*(n+1)*(11*n +10)/6.

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A177254 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} having k adjacent blocks (0 <= k <= n). An adjacent block is a block of the form (i, i+1, i+2, ...).

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A190048 Expansion of (8+6*x)/(1-x)^5.

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Magma

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PARI

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A190049 Expansion of (16+24*x+2*x^2)/(x-1)^6.

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Magma

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PARI

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A126264 a(n) = 5*n^2 + 3*n.

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A254407 a(n) = n(n+1)(11*n +10)/6.

A190049 Expansion of (16+24x+2x^2)/(x-1)^6.

A126264 a(n) = 5n^2 + 3n.