A055584 -id:A055584 - cp's OEIS frontend

A055585 Second column of triangle A055584.

1, 6, 25, 88, 280, 832, 2352, 6400, 16896, 43520, 109824, 272384, 665600, 1605632, 3829760, 9043968, 21168128, 49152000, 113311744, 259522560, 590872576, 1337982976, 3014656000, 6761218048, 15099494400, 33587986432, 74440507392

Offset: 0

Wolfdieter Lang, May 26 2000

Number of 132-avoiding permutations of [n+5] containing exactly three 123 patterns. - Emeric Deutsch, Jul 13 2001
If X_1,X_2,...,X_n are 2-blocks of a (2n+2)-set X then, for n>=1, a(n-1) is the number of (n+3)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 18 2007
Convolution of A001792 with itself. - Philippe Deléham, Feb 21 2013

			a(1)=6 because 432516,432561,435126,452136,532146 and 632145 are the only 132-avoiding permutations of 123456, containing exactly three increasing subsequences of length 3.

Milan Janjic, Two Enumerative Functions
M. Janjic, On a class of polynomials with integer coefficients, JIS 11 (2008) 08.5.2
Pudwell, Lara; Scholten, Connor; Schrock, Tyler; Serrato, Alexa Noncontiguous pattern containment in binary trees, ISRN Comb. 2014, Article ID 316535, 8 p. (2014), Section 5.2.
A. Robertson, H. S. Wilf and D. Zeilberger, Permutation patterns and continued fractions, Electr. J. Combin. 6, 1999, #R38.
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Cf. A055584, partial sums of A049612, n >= 1.

Table[(1/3)*2^(n-3)*(n+1)*(n+3)*(n+8), {n,0,50}] (* G. C. Greubel, Aug 22 2015 *)
LinearRecurrence[{8,-24,32,-16},{1,6,25,88},30] (* Harvey P. Dale, Nov 03 2017 *)

Vec(((1-x)^2)/(1-2*x)^4 + O(x^30)) \\ Michel Marcus, Aug 22 2015

G.f.: (1-x)^2/(1-2*x)^4.
a(n) = A055584(n+1, 1). a(n) = sum(a(j), j=0..n-1)+A001793(n+1), n >= 1.
a(n) = 2^(n-3)(n+1)(n+3)(n+8)/3.
Preceded by 0, this is the binomial transform of the tetrahedral numbers A000292. - Carl Najafi, Sep 08 2011
E.g.f.: (1/6)*(2*x^3 + 15*x^2 + 24*x + 6)*exp(2*x). - G. C. Greubel, Aug 22 2015

A055586 Sixth column of triangle A055584.

Original entry on oeis.org

1, 10, 59, 268, 1037, 3598, 11535, 34832, 100369, 278546, 749587, 1966100, 5046293, 12714006, 31522839, 77070360, 186122265, 444596250, 1051721755, 2466250780, 5737807901, 13254000670, 30417092639, 69390565408

Offset: 0

Wolfdieter Lang, May 26 2000

A. F. Y. Zhao, Pattern Popularity in Multiply Restricted Permutations, Journal of Integer Sequences, 17 (2014), #14.10.3.
Index entries for linear recurrences with constant coefficients, signature (10,-41,88,-104,64,-16).

Cf. A055584.

Partial sums of A027608.

CoefficientList[Series[1/(((1-x)^2)(1-2x)^4),{x,0,30}],x] (* or *) LinearRecurrence[{10,-41,88,-104,64,-16},{1,10,59,268,1037,3598},30] (* Harvey P. Dale, Jul 31 2025 *)

Vec(1/(((1-x)^2)*(1-2*x)^4) + O(x^40)) \\ Michel Marcus, Dec 11 2015

G.f.: 1/(((1-x)^2)*(1-2*x)^4).
a(n) = A055584(n+5, 5).
a(n) = Sum_{j=0..n-1} a(j) + A055582(n) for n >= 1.
E.g.f.: exp(x)*(x + 9) + 8*exp(2*x)*(2*x^3 + 3*x^2 + 6*x - 3)/3. - Stefano Spezia, Sep 24 2024

A049612 a(n) = T(n,3), array T as in A049600.

Original entry on oeis.org

0, 1, 5, 19, 63, 192, 552, 1520, 4048, 10496, 26624, 66304, 162560, 393216, 940032, 2224128, 5214208, 12124160, 27983872, 64159744, 146210816, 331350016, 747110400, 1676673024, 3746562048, 8338276352, 18488492032

Offset: 0

Clark Kimberling

nonn

If X_1, X_2, ..., X_n are 2-blocks of a (2n+3)-set X then, for n >= 1, a(n+1) is the number of (n+3)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 18 2007

Robert Cori, Gabor Hetyei, Genus one partitions, in 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), 2014, Chicago, United States. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AT, pp. 333-344,

R. Cori and G. Hetyei, Counting genus one partitions and permutations, arXiv preprint arXiv:1306.4628 [math.CO], 2013.
R. Cori and G. Hetyei, How to count genus one partitions, FPSAC 2014, Chicago, Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France, 2014, 333-344.
Milan Janjic, Two Enumerative Functions
S. Kitaev and J. Remmel, p-Ascent Sequences, arXiv preprint arXiv:1503.00914 [math.CO], 2015.
Sergey Kitaev and Jeffrey Remmel, A note on p-ascent sequences, preprint, 2016.
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16)

Cf. A049600.

Row sums of triangle A055252. a(n+1) = A055584(n, 0), n >= 0.

CoefficientList[Series[x (1-x)^3/(1-2x)^4,{x,0,30}],x] (* or *) Join[ {0},LinearRecurrence[{8,-24,32,-16},{1,5,19,63},30]] (* Harvey P. Dale, Jan 07 2014 *)

G.f.: x*(1-x)^3 /(1-2*x)^4.
a(n) = Sum_{k=0..floor((n+3)/2)} C(n+3, 2k)*C(k+1, 3). - Paul Barry, May 15 2003
a(n+1) = 2^n*n^3/48 + 5*2^n*n^2/16 + 7*2^n*n/6 + 2^n, n>=1. - Milan Janjic, Nov 18 2007
Binomial transform of the tetrahedral numbers A000292 when omitting the initial 0 in both sequences. - Carl Najafi, Sep 08 2011

A055252 Triangle of partial row sums (prs) of triangle A055249.

Original entry on oeis.org

1, 4, 1, 13, 5, 1, 38, 18, 6, 1, 104, 56, 24, 7, 1, 272, 160, 80, 31, 8, 1, 688, 432, 240, 111, 39, 9, 1, 1696, 1120, 672, 351, 150, 48, 10, 1, 4096, 2816, 1792, 1023, 501, 198, 58, 11, 1, 9728, 6912, 4608, 2815, 1524, 699, 256, 69, 12, 1, 22784, 16640, 11520

Offset: 0

Wolfdieter Lang, May 26 2000

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is (((1-z)^2)/(1-2*z)^3)/(1-x*z/(1-z)).
This is the third member of the family of Riordan-type matrices obtained from A007318(n,m) (Pascal's triangle read as lower triangular matrix) by repeated application of the prs-procedure.
The column sequences appear as A049611(n+1), A001793, A001788, A055580, A055581, A055582, A055583 for m=0..6.

			[0] 1
[1] 4, 1
[2] 13, 5, 1
[3] 38, 18, 6, 1
[4] 104, 56, 24, 7, 1
[5] 272, 160, 80, 31, 8, 1
[6] 688, 432, 240, 111, 39, 9, 1
[7] 1696, 1120, 672, 351, 150, 48, 10, 1
Fourth row polynomial (n = 3): p(3, x) = 38 + 18*x + 6*x^2 + x^3.

Cf. A007318, A055248, A055249. Row sums: A049612(n+1)= A055584(n, 0).

T := (n, k) -> binomial(n, k)*hypergeom([3, k - n], [k + 1], -1):
for n from 0 to 7 do seq(simplify(T(n, k)), k = 0..n) od; # Peter Luschny, Sep 23 2024

a(n, m)=sum(A055249(n, k), k=m..n), n >= m >= 0, a(n, m) := 0 if n

Column m recursion: a(n, m)= sum(a(j, m), j=m..n-1)+ A055249(n, m), n >= m >= 0, a(n, m) := 0 if n

G.f. for column m: (((1-x)^2)/(1-2*x)^3)*(x/(1-x))^m, m >= 0.

T(n, k) = binomial(n, k)*hypergeom([3, k - n], [k + 1], -1). - Peter Luschny, Sep 23 2024

A055587 Triangle with columns built from row sums of the partial row sums triangles obtained from Pascal's triangle A007318. Essentially A049600 formatted differently.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 8, 4, 1, 1, 16, 20, 13, 5, 1, 1, 32, 48, 38, 19, 6, 1, 1, 64, 112, 104, 63, 26, 7, 1, 1, 128, 256, 272, 192, 96, 34, 8, 1, 1, 256, 576, 688, 552, 321, 138, 43, 9, 1, 1, 512, 1280, 1696, 1520, 1002, 501, 190, 53, 10, 1, 1, 1024, 2816, 4096

Offset: 0

Wolfdieter Lang, May 30 2000

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is 1/((1-z)*(1-x*z*(1-z)/(1-2*z))).

Column m (without leading zeros) is obtained from convolution of A000012 (powers of 1) with m-fold convoluted A011782.

			{1}; {1, 1}; {1, 2, 1}; {1, 4, 3, 1}; {1, 8, 8, 4, 1}; ...
Fourth row polynomial (n=3): p(3,x)= 1+4*x+3*x^2+x^3

Cf. A049600, column sequences are A000012 (powers of 1), A000079 (powers of 2), A001792, A049611, A049612, A055589, A055852-5 for m=0..9, row sums: A055588.

t[n_, k_] := Hypergeometric2F1[k, k-n, 1, -1]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 05 2014, after Paul D. Hanna *)

{T(n,k) = if( n<0 || k<0, 0, polcoeff( polcoeff( 1 / ((1 - z) * (1 - x*z * (1 - z) / (1 - 2*z) + z * O(z^n) + x * O(x^k))), k), n))}; /* Michael Somos, Sep 30 2003 */

{T(n,k)=if(k>n||n<0||k<0,0,if(k==0||k==n,1, sum(j=0,n-k,binomial(n-k,j)*binomial(k+j-1,k-1)););)} (Hanna)

a(n, m)= Am(n, 0) if n >= m >= 0 and a(n, m) := 0 if nA007318) with the lower triangular matrix A007318 (Pascal triangle) and prs^(m) is the partial row sums (prs) mapping for triangular matrices applied m times. See e.g. A055584 for m=4.
G.f. for column m: (1/(1-x))*(x*(1-x)/(1-2*x))^m, m >= 0.
T(n, k) = sum_{j=0..n-k} C(n-k, j)*C(k+j-1, k-1). - Paul D. Hanna, Jan 14 2004

A055855 Convolution of A055854 with A011782.

Original entry on oeis.org

0, 1, 10, 64, 328, 1462, 5908, 22180, 78592, 265729, 864146, 2719028, 8316200, 24814832, 72453344, 207502016, 584094080, 1618757120, 4423347200, 11932579840, 31812874240, 83901227008, 219074805760, 566754967552

Offset: 0

Wolfdieter Lang May 30 2000

Tenth column of triangle A055587.

T(n,8) of array T as in A049600.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (18,-144,672,-2016,4032,-5376,4608,-2304,512).

Cf. A011782, A049600, A055584, A055857.

R:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( x*(1-x)^8/(1-2*x)^9 )); // G. C. Greubel, Jan 16 2020

seq(coeff(series(x*(1-x)^8/(1-2*x)^9, x, n+1), x, n), n = 0..30); # G. C. Greubel, Jan 16 2020

CoefficientList[Series[x*(1-x)^8/(1-2*x)^9, {x,0,30}], x] (* G. C. Greubel, Jan 16 2020 *)

my(x='x+O('x^30)); concat([0], Vec(x*(1-x)^8/(1-2*x)^9)) \\ G. C. Greubel, Jan 16 2020

def A055855_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1-x)^8/(1-2*x)^9 ).list()
A055855_list(30) # G. C. Greubel, Jan 16 2020

a(n) = T(n, 8) = A055587(n+8, 9).

G.f.: x*(1-x)^8/(1-2*x)^9.

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A055585 Second column of triangle A055584.

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A055586 Sixth column of triangle A055584.

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A049612 a(n) = T(n,3), array T as in A049600.

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A055252 Triangle of partial row sums (prs) of triangle A055249.

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A055587 Triangle with columns built from row sums of the partial row sums triangles obtained from Pascal's triangle A007318. Essentially A049600 formatted differently.

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A055855 Convolution of A055854 with A011782.

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