A079309 -id:A079309 - cp's OEIS frontend

A120279 a(n) = Sum[Sum[(i+j)!/i!/j!,{i,1,j}],{j,1,n}].

2, 11, 45, 170, 631, 2346, 8780, 33089, 125466, 478181, 1830258, 7030557, 27088856, 104647615, 405187809, 1571990918, 6109558567, 23782190466, 92705454875, 361834392094, 1413883873953, 5530599237752, 21654401079301, 84859704298176

Offset: 1

Alexander Adamchuk, Jul 05 2006

p divides a(p-1) and a(p-2) for prime p=5,11,17,23,29,41,47,53,59,71..=A007528[n] Primes of form 6n-1.
p divides a([(2p-1)/2]) for prime p=5,11,17,23,29,41,47,53,59,71..=A007528[n] Primes of form 6n-1.
p divides a((p-5)/2) for prime p=17,29,41,53,89,101.. =A040115[n] Primes of form 12n+5. Primes congruent to 5 (mod 12) excluding 5.
p divides a((p-5)/3) for prime p=11,17,23,29,41,47,53,59,71..=A007528[n] Primes of form 6n-1 excluding 5.
p divides a([(p-3)/3]) for prime p=11,17,23,29,41,47,53,59,71..=A007528[n] Primes of form 6n-1 excluding 5.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A007528, A007528, A040115, A048775, A079309, A079309, A066796.

Table[Sum[Sum[(i+j)!/i!/j!,{i,1,j}],{j,1,n}],{n,1,50}]

a(n) = Sum[Sum[(i+j)!/i!/j!,{i,1,j}],{j,1,n}]. a(n) = A079309(n+1) - (n+1). a(n) = A066796(n+1)/2 - (n+1).
Recurrence: (n+1)*(3*n-2)*a(n) = 6*(3*n^2-1)*a(n-1) - 3*(9*n^2-n-2)*a(n-2) + 2*(2*n-1)*(3*n+1)*a(n-3). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 2^(2*n+3)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 19 2012
a(n) = Sum_{k=1..n} Sum_{i=1..k} C(k+i,i). - Wesley Ivan Hurt, Sep 19 2017

A138341 Expansion of (1-4x-x^3)/(1-x+x^2)^2.

Original entry on oeis.org

1, -2, -7, -7, 2, 13, 13, -2, -19, -19, 2, 25, 25, -2, -31, -31, 2, 37, 37, -2, -43, -43, 2, 49, 49, -2, -55, -55, 2, 61, 61, -2, -67, -67, 2, 73, 73, -2, -79, -79, 2, 85, 85, -2, -91, -91, 2, 97, 97, -2, -103, -103, 2, 109, 109, -2, -115, -115, 2, 121, 121, -2

Offset: 0

Paul Barry, Mar 15 2008

Hankel transform of A079309.

Index entries for linear recurrences with constant coefficients, signature (2,-3,2,-1).

Cf. A079309, A138342.

CoefficientList[Series[(1-4x-x^3)/(1-x+x^2)^2,{x,0,100}],x] (* or *) LinearRecurrence[{2,-3,2,-1},{1,-2,-7,-7},100] (* Harvey P. Dale, Sep 23 2021 *)

Vec((1-4*x-x^3)/(1-x+x^2)^2 + O(x^62)) \\ Jinyuan Wang, Apr 09 2020

a(n) = (2n+1)*cos(Pi*n/3) - (2n+5)*sin(Pi*n/3)/sqrt(3).

a(n) = 2*a(n-1) - 3*a(n-2) + 2*a(n-3) - a(n-4) for n > 3. - Jinyuan Wang, Apr 09 2020

More terms from Jinyuan Wang, Apr 09 2020

A143952 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k peak plateaux (0<=k<=floor(n/2)). A peak plateau is a run of consecutive peaks that is preceded by an upstep and followed by a down step; a peak consists of an upstep followed by a downstep.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 12, 1, 1, 33, 8, 1, 88, 42, 1, 1, 232, 183, 13, 1, 609, 717, 102, 1, 1, 1596, 2622, 624, 19, 1, 4180, 9134, 3275, 205, 1, 1, 10945, 30691, 15473, 1650, 26, 1, 28656, 100284, 67684, 11020, 366, 1, 1, 75024, 320466, 279106, 64553, 3716, 34, 1

Offset: 0

Emeric Deutsch, Oct 10 2008

Row n has 1+floor(n/2) terms.
Row sums are the Catalan numbers (A000108).
T(n,1)=A027941(n-1)=Fibonacci(2n-1)-1.
Sum(k*T(n,k),k=0..floor(n/2))=A079309(n-1).
For the statistic "number of peaks in peak plateaux", see A143953.

			T(3,1)=4 because we have UD(UUDD), (UUDD)UD, (UUDUDD) and U(UUDD)D (the peak plateaux are shown between parentheses).
The triangle starts:
1;
1;
1,1;
1,4;
1,12,1;
1,33,8;
1,88,42,1;

David Callan, Kreweras's Narayana Number Identity Has a Simple Dyck Path Interpretation .

Cf. A000045, A000108, A027941, A079309, A143953.

C:=proc(z) options operator, arrow: (1/2-(1/2)*sqrt(1-4*z))/z end proc: G:=(1-z)*C(z*(1-z)^2/(1-z+z^2-t*z^2)^2)/(1-z+z^2-t*z^2): Gser:=simplify(series(G,z= 0,17)): for n from 0 to 14 do P[n]:=sort(coeff(Gser,z,n)) end do: for n from 0 to 14 do seq(coeff(P[n],t,j),j=0..floor((1/2)*n)) end do; # yields sequence in triangular form

The g.f. G=G(t,z) satisfies z(1-z)G^2 - (1-z+z^2-tz^2)G+1-z = 0 (for the explicit form of G see the Maple program).
The trivariate g.f. g=g(x,y,z) of Dyck paths with respect to number of peak plateaux, number of peaks in the peak plateaux and semilength, marked, by x, y and z, respectively satisfies g=1+zg[g+xyz/(1-yz)-z/(1-z)].
T(n,k) = Sum_{r=1..n} Narayana(n-r,k)*binomial(2n-r-k,r-k) where Narayana(n,k) := binomial(n,k)*binomial(n,k-1)/n is the Narayana number A001263. - David Callan, Oct 31 2008

A143954 Number of peaks in the peak plateaux of all Dyck paths of semilength n.

Original entry on oeis.org

0, 0, 1, 5, 19, 68, 243, 880, 3233, 12021, 45119, 170595, 648787, 2479057, 9509627, 36598497, 141246127, 546433952, 2118424887, 8227983472, 32010173957, 124715628852, 486550020967, 1900433894942, 7431033132717, 29085434212042

Offset: 0

Emeric Deutsch, Oct 10 2008

nonn

A peak plateau is a run of consecutive peaks that is preceded by an upstep and followed by a down step; a peak consists of an upstep followed by a downstep.

			a(3)=5 because in the peak plateaux of the Dyck paths UDUDUD, UD(UUDD), (UUDD)UD, (UUDUDD) and U(UUDD)D, shown between parentheses, we have 0 + 1 + 1 + 2 + 1 = 5 peaks.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A143952, A143953, A079309.

C:=((1-sqrt(1-4*z))*1/2)/z: G:=z^2*C/((1-z)^2*sqrt(1-4*z)): Gser:=series(G,z= 0,30): seq(coeff(Gser,z,n),n=0..25);

CoefficientList[Series[x^2*((1-Sqrt[1-4*x])*1/2)/x/((1-x)^2*Sqrt[1-4*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

x='x+O('x^50); concat([0,0], Vec(x*(1-sqrt(1-4*x))/(2*(1-x)^2*sqrt(1-4*x)))) \\ G. C. Greubel, Mar 22 2017

a(n) = Sum_{k=0..n-1} k*A143953(n,k).
G.f.: z^2*C/[(1-z)^2*sqrt(1-4z)], where C = [1-sqrt(1-4z)]/(2z) is the Catalan function.
a(n) ~ 2^(2*n+1)/(9*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014
a(n) = Sum_{k=1..n-1} A079309(k). - Doug Bell, Jun 23 2015
Conjecture: (-n+1)*a(n) +2*(3*n-4)*a(n-1) +(-9*n+13)*a(n-2) +2*(2*n-3)*a(n-3)=0. - R. J. Mathar, Jun 16 2016

A361653 Number of even-length integer partitions of n with integer median.

Original entry on oeis.org

0, 0, 1, 0, 3, 1, 5, 3, 11, 7, 17, 16, 32, 31, 52, 55, 90, 99, 144, 167, 236, 273, 371, 442, 587, 696, 901, 1078, 1379, 1651, 2074, 2489, 3102, 3707, 4571, 5467, 6692, 7982, 9696, 11543, 13949, 16563, 19891, 23572, 28185, 33299, 39640, 46737, 55418, 65164

Offset: 0

Gus Wiseman, Mar 23 2023

nonn

The median of an even-length multiset is the average of the two middle parts.

Because any odd-length partition has integer median, the odd-length version is counted by A027193, strict case A067659.

			The a(2) = 1 through a(9) = 7 partitions:
  (11)  .  (22)    (2111)  (33)      (2221)    (44)        (3222)
           (31)            (42)      (4111)    (53)        (4221)
           (1111)          (51)      (211111)  (62)        (4311)
                           (3111)              (71)        (6111)
                           (111111)            (2222)      (321111)
                                               (3221)      (411111)
                                               (3311)      (21111111)
                                               (5111)
                                               (221111)
                                               (311111)
                                               (11111111)
For example, the partition (4,3,1,1) has length 4 and median 2, so is counted under a(9).

The odd-length version is counted by A027193, strict A067659.
Including odd-length partitions gives A307683, complement A325347.
For mean instead of median we have A361655, any length A067538.
A000041 counts integer partitions, strict A000009.
A000975 counts subsets with integer median, mean A051293.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A008284, A013580, A079309, A240219, A240850, A349156, A359897, A359908, A359912, A360005, A360952.

Table[Length[Select[IntegerPartitions[n], EvenQ[Length[#]]&&IntegerQ[Median[#]]&]],{n,0,30}]

A363223 Numbers with bigomega equal to median prime index.

Original entry on oeis.org

2, 9, 10, 50, 70, 75, 105, 110, 125, 130, 165, 170, 175, 190, 195, 230, 255, 275, 285, 290, 310, 325, 345, 370, 410, 425, 430, 435, 465, 470, 475, 530, 555, 575, 590, 610, 615, 645, 670, 686, 705, 710, 725, 730, 775, 790, 795, 830, 885, 890, 915, 925, 970

Offset: 1

Gus Wiseman, May 29 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
    2: {1}
    9: {2,2}
   10: {1,3}
   50: {1,3,3}
   70: {1,3,4}
   75: {2,3,3}
  105: {2,3,4}
  110: {1,3,5}
  125: {3,3,3}
  130: {1,3,6}
  165: {2,3,5}
  170: {1,3,7}
  175: {3,3,4}

For maximum instead of median we have A106529, counted by A047993.
For minimum instead of median we have A324522, counted by A006141.
Partitions of this type are counted by A361800.
For twice median we have A362050, counted by A362049.
For maximum instead of length we have A362621, counted by A053263.
A000975 counts subsets with integer median.
A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
A359908 lists numbers whose prime indices have integer median.
A360005 gives twice median of prime indices.
Cf. A000040, A013580, A079309, A240219, A327473, A327476, A361860, A362619, A362622, A362980.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],PrimeOmega[#]==Median[prix[#]]&]

2*A001222(a(n)) = A360005(a(n)).

A120278 a(n) = Sum_{m=1..n} Sum_{k=1..m} C(2k,k), where C(2k,k) = (2*k)!/(k!)^2 = A000984(k).

Original entry on oeis.org

2, 10, 38, 136, 486, 1760, 6466, 24042, 90238, 341190, 1297574, 4958114, 19019254, 73196994, 282492254, 1092867904, 4236849774, 16455966944, 64020347914, 249431257704, 973100041934, 3800867789884, 14862066265434, 58170868424084

Offset: 1

Alexander Adamchuk, Jul 04 2006

a(2*(p-1)) is divisible by p^2 for p=7,13,19,31,37,43,61,67.. A002476 (Primes of the form 6m + 1).

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A000984, A066796, A002476.

Cf. A066796, A079309.

Table[Sum[Sum[(2k)!/(k!)^2,{k,1,m}],{m,1,n}],{n,1,50}]
CoefficientList[Series[(1/Sqrt[1-4 x]-1)/((x-1)^2 x),{x,0,50}],x] (* Harvey P. Dale, May 24 2011 *)

a(n) = Sum_{m=1..n} Sum_{k=1..m} (2*k)!/(k!)^2.
a(n) = 2 * Sum_{k=1..n} A079309(k) = Sum_{k=1..n} A066796(k). - Alexander Adamchuk, Sep 01 2006
G.f.: x*(1/sqrt(1-4*x)-1)/(x*(x-1)^2). - Harvey P. Dale, May 24 2011
Recurrence: n*a(n) = 2*(3*n-1)*a(n-1) - (9*n-4)*a(n-2) + 2*(2*n-1)*a(n-3). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 2^(2*n+4)/(9*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 19 2012

A361863 Number of set partitions of {1..n} such that the median of medians of the blocks is (n+1)/2.

Original entry on oeis.org

1, 2, 3, 9, 26, 69, 335, 1018, 6629, 22805, 182988, 703745

Offset: 1

Gus Wiseman, Apr 04 2023

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Since (n+1)/2 is the median of {1..n}, this sequence counts "transitive" set partitions.

			The a(1) = 1 through a(4) = 9 set partitions:
  {{1}}  {{12}}    {{123}}      {{1234}}
         {{1}{2}}  {{13}{2}}    {{12}{34}}
                   {{1}{2}{3}}  {{124}{3}}
                                {{13}{24}}
                                {{134}{2}}
                                {{14}{23}}
                                {{1}{23}{4}}
                                {{14}{2}{3}}
                                {{1}{2}{3}{4}}
The set partition {{1,4},{2,3}} has medians {5/2,5/2}, with median 5/2, so is counted under a(4).
The set partition {{1,3},{2,4}} has medians {2,3}, with median 5/2, so is counted under a(4).

For mean instead of median we have A361910.
A000110 counts set partitions.
A000975 counts subsets with integer median, mean A327475.
A013580 appears to count subsets by median, A327481 by mean.
A325347 counts partitions w/ integer median, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
A360005 gives twice median of prime indices, distinct A360457.
A361864 counts set partitions with integer median of medians, means A361865.
A361866 counts set partitions with integer sum of medians, means A361911.
Cf. A007837, A035470, A038041, A275714, A275780, A326512, A326513.
Cf. A027193, A067659, A079309, A231147, A326516, A326521, A326537, A359907, A361654, A361801.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],(n+1)/2==Median[Median/@#]&]],{n,6}]

A364026 Table read by descending antidiagonals. T(n,k) is the big Ramsey degree of k in w^n, where w is the first transfinite ordinal, omega.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 4, 1, 1, 0, 1, 26, 14, 1, 1, 0, 1, 236, 509, 49, 1, 1, 0, 1, 2752, 35839, 10340, 175, 1, 1, 0, 1, 39208, 4154652, 5941404, 222244, 637, 1, 1, 0, 1, 660032, 718142257, 7244337796, 1081112575, 4981531, 2353, 1, 1, 0, 1, 12818912, 173201493539

Offset: 0

Nathan Hurtig, Jul 01 2023

T(n,k) is the least integer t such that, for all finite colorings of the k-subsets of w^n, there exists some S, an order-equivalent subset to w^n, where that coloring restricted to the k-subsets of S outputs at most t colors.
By Ramsey's theorem, the first row T(1,k)=1 for all k.
The second row T(2,k) coincides with A000311.
The second column T(n,2) coincides with A079309.

			The data is organized in a table beginning with row n = 0 and column k = 0. The data is read by descending antidiagonals. T(2,3)=26.
The table T(n,k) begins:
[n/k]   0   1      2        3       4         5   ...
--------------------------------------------------------------------
[0]     1,  1,     0,       0,      0,        0,  ...
[1]     1,  1,     1,       1,      1,        1,  ...
[2]     1,  1,     4,      26,    236,     2572,  ...
[3]     1,  1,    14,     509,  35839,  4154652,  ...
[4]     1,  1,    49,   10340,  ...
[5]     1,  1,   175,  222244,  ...
[6]     1,  1,   637,  ...

Dragan Mašulovic and Branislav Šobot, Countable ordinals and big Ramsey degrees, Combinatorica, 41 (2021), 425-446.
Alexander S. Kechris, Vladimir G. Pestov, and Stevo Todorčević, Fraïssé Limits, Ramsey Theory, and topological dynamics of automorphism groups, Geometric & Functional Analysis, 15 (2005), 106-189.

Joanna Boyland, William Gasarch, Nathan Hurtig, and Robert Rust, Big Ramsey Degrees of Countable Ordinals, arXiv:2305.07192 [math.CO], 2023.
Eric Weisstein's World of Mathematics, Ordinal Number.

T(2,k) is A000311. T(n,2) is A079309.

pp p n k
  | n == 0 && k >= 2 = 0
  | k == 0 && p == 0 = 1
  | k == 0 && p >= 1 = 0
  | n == 0 && k == 1 && p == 0 = 1
  | n == 0 && k == 1 && p >= 1 = 0
  | n == 1 && k >= 1 && k == p = 1
  | n == 1 && k >= 1 && k /= p = 0
  | n >= 2 && k >= 1 = sum [binom (p-1) i * pp i (n-1) j * pp (p-1-i) n (k-j) | i <- [0..p-1], j <- [1..k]]
binom n 0 = 1
binom 0 k = 0
binom n k = binom (n-1) (k-1) * n `div` k
a364026 n k =
  sum [pp p n k | p <- [0..n*k]]

T(n,k) = Sum_{p=0..n*k} P(p,n,k), where for n >= 2 and k >= 1,
P(0,n,k) = 0, and for p >= 1,
P(p,n,k) = Sum_{j=1..k} Sum_{0..p-1} binomial(p-1,i)*P(i,n-1,j)*P(p-1-i,n,k-j).

A361802 Irregular triangle read by rows where T(n,k) is the number of k-subsets of {-n+1,...,n} with sum 0, for k = 1,...,2n-1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 3, 6, 7, 5, 2, 1, 1, 4, 10, 16, 18, 14, 8, 3, 1, 1, 5, 15, 31, 46, 51, 43, 27, 12, 3, 1, 1, 6, 21, 53, 98, 139, 155, 134, 88, 43, 16, 4, 1, 1, 7, 28, 83, 184, 319, 441, 486, 424, 293, 161, 68, 21, 4, 1

Offset: 1

Gus Wiseman, Apr 10 2023

Also the number of k-subsets of {1,...,2n} with mean n.

			Triangle begins:
   1
   1   1   1
   1   2   3   2   1
   1   3   6   7   5   2   1
   1   4  10  16  18  14   8   3   1
   1   5  15  31  46  51  43  27  12   3   1
   1   6  21  53  98 139 155 134  88  43  16   4   1
   1   7  28  83 184 319 441 486 424 293 161  68  21   4   1
Row n = 3 counts the following subsets:
  {0}  {-1,1}  {-1,0,1}   {-2,-1,0,3}  {-2,-1,0,1,2}
       {-2,2}  {-2,0,2}   {-2,-1,1,2}
               {-2,-1,3}

Row lengths are A005408.
Row sums are A212352.
A007318 counts subsets by length.
A067538 counts partitions with integer mean.
A231147 counts subsets by median.
A327475 counts subsets with integer mean, median A000975.
A327481 counts subsets by mean.
Cf. A006134, A013580, A024718, A079309, A326512, A349156, A361654, A362046.

Table[Length[Select[Subsets[Range[-n+1,n],{k}],Total[#]==0&]],{n,6},{k,2n-1}]

cp's OEIS Frontend

A120279 a(n) = Sum[Sum[(i+j)!/i!/j!,{i,1,j}],{j,1,n}].

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A138341 Expansion of (1-4x-x^3)/(1-x+x^2)^2.

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A143952 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k peak plateaux (0<=k<=floor(n/2)). A peak plateau is a run of consecutive peaks that is preceded by an upstep and followed by a down step; a peak consists of an upstep followed by a downstep.

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Maple

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A143954 Number of peaks in the peak plateaux of all Dyck paths of semilength n.

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Maple

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A361653 Number of even-length integer partitions of n with integer median.

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A363223 Numbers with bigomega equal to median prime index.

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A120278 a(n) = Sum_{m=1..n} Sum_{k=1..m} C(2*k,k), where C(2*k,k) = (2*k)!/(k!)^2 = A000984(k).

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A361863 Number of set partitions of {1..n} such that the median of medians of the blocks is (n+1)/2.

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A120278 a(n) = Sum_{m=1..n} Sum_{k=1..m} C(2k,k), where C(2k,k) = (2*k)!/(k!)^2 = A000984(k).