A038165 -id:A038165 - cp's OEIS frontend

A059594 Convolution triangle based on A008619 (positive integers repeated).

1, 1, 1, 2, 2, 1, 2, 5, 3, 1, 3, 8, 9, 4, 1, 3, 14, 19, 14, 5, 1, 4, 20, 39, 36, 20, 6, 1, 4, 30, 69, 85, 60, 27, 7, 1, 5, 40, 119, 176, 160, 92, 35, 8, 1, 5, 55, 189, 344, 376, 273, 133, 44, 9, 1, 6, 70, 294, 624, 820, 714, 434

Offset: 0

Wolfdieter Lang, Feb 02 2001

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group.
The G.f. for the row polynomials p(n,x) = Sum_{m=0..n} a(n,m)*x^m is 1/((1-z^2)*(1-z)-x*z).
The column sequences are A008619(n); A006918(n); A038163(n-2), n >= 2; A038164(n-3), n >= 3; A038165(n-4), n >= 4; A038166(n-5), n >= 5; A059595(n-6), n >= 6; A059596(n-7), n >= 7; A059597(n-8), n >= 8; A059598(n-9), n >= 9; A059625(n-10), n >= 10 for m=0..10.
The sequence of row sums is A006054(n+2).
From Gary W. Adamson, Aug 14 2016: (Start)
The sequence can be generated by extracting the descending antidiagonals of an array formed by taking powers of the natural integers with repeats, (1, 1, 2, 2, 3, 3, ...), as follows:
  1, 1,  2,  2,  3,   3, ...
  1, 2,  5,  8, 14,  20, ...
  1, 3,  9, 19, 39,  69, ...
  1, 4, 14, 36, 85, 176, ...
  ...
Row sums of the triangle = (1, 2, 5, 11, 25, 56, ...), the INVERT transform of (1, 1, 2, 2, 3, 3, ...). (End)

			{1}; {1,1}; {2,2,1}; {2,5,3,1}; ...
Fourth row polynomial (n=3): p(3,x)= 2 + 5*x + 3*x^2 + x^3.

t[n_, m_] := Sum[Sum[Binomial[j, n-m-3*k+2*j]*(-1)^(j-k)*Binomial[k, j], {j, 0, k}]*Binomial[m+k, m], {k, 0, n-m}]; Table[t[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, May 27 2013, after Vladimir Kruchinin *)

T(n,m):=sum((sum(binomial(j,n-m-3*k+2*j)*(-1)^(j-k)*binomial(k,j),j,0,k)) *binomial(m+k,m),k,0,n-m); /* Vladimir Kruchinin, Dec 14 2011 */

a(n, m) := a(n-1, m) + (-(n-m+1)*a(n, m-1) + 3*(n+2*m)*a(n-1, m-1))/(8*m), n >= m >= 1; a(n, 0) := floor((n+2)/2) = A008619(n), n >= 0; a(n, m) := 0 if n < m.
G.f.for column m >= 0: ((x/((1-x^2)*(1-x)))^m)/((1-x^2)*(1-x)).
T(n,m) = Sum_{k=0..n-m} (Sum_{j=0..k} binomial(j, n-m-3*k+2*j)*(-1)^(j-k)*binomial(k,j))*binomial(m+k,m). - Vladimir Kruchinin, Dec 14 2011
Recurrence: T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-3,k) with T(0,0) = 1. - Philippe Deléham, Feb 23 2012

A038166 G.f.: 1/((1-x)*(1-x^2))^6.

Original entry on oeis.org

1, 6, 27, 92, 273, 714, 1715, 3816, 8007, 15938, 30381, 55692, 98735, 169806, 284349, 464672, 742950, 1164228, 1791426, 2710344, 4037670, 5928988, 8591154, 12294672, 17392258, 24337404, 33711510, 46251016, 62886162, 84779748

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (6, -9, -16, 60, -24, -116, 144, 66, -220, 66, 144, -116, -24, 60, -16, -9, 6, -1).

Cf. A008619, A006918, A038163-A038165.

A038166 := proc(n)
    add( A038163(n-i)*A038163(i),i=0..n) ;
end proc:
seq(A038166(n),n=0..30) ;# R. J. Mathar, Feb 22 2021

CoefficientList[Series[1/((1-x)(1-x^2))^6,{x,0,40}],x] (* or *) LinearRecurrence[ {6,-9,-16,60,-24,-116,144,66,-220,66,144,-116,-24,60,-16,-9,6,-1},{1,6,27,92,273,714,1715,3816,8007,15938,30381,55692,98735,169806,284349,464672,742950,1164228},40] (* Harvey P. Dale, Jun 10 2013 *)

a(2*k) = binomial(k + 8, 8)*(2*k + 9)*(8*k^2 + 72*k + 55)/(11*5*9) = A059603(k); a(2*k + 1) = 2*binomial(k + 9, 9)*(8*k^2 + 80*k + 165)/(11*5) = 2*A059624(k), k >= 0; Wolfdieter Lang, Feb 02 2000

a(0)=1, a(1)=6, a(2)=27, a(3)=92, a(4)=273, a(5)=714, a(6)=1715, a(7)=3816, a(8)=8007, a(9)=15938, a(10)=30381, a(11)=55692, a(12)=98735, a(13)=169806, a(14)=284349, a(15)=464672, a(16)=742950, a(17)=1164228, a(n)=6*a(n-1)-9*a(n-2)-16*a(n-3)+60*a(n-4)-24*a(n-5)-116*a(n-6)+144*a(n-7)+ 66*a(n-8)- 220*a(n-9)+66*a(n-10)+144*a(n-11)-116*a(n-12)-24*a(n-13)+60*a(n-14)-16*a(n-15)- 9*a(n-16)+ 6*a(n-17)-a(n-18). - Harvey P. Dale, Jun 10 2013

A060101 Sixth column (m=5) of triangle A060098.

Original entry on oeis.org

1, 6, 26, 86, 246, 622, 1442, 3102, 6292, 12122, 22374, 39754, 68354, 114114, 185614, 294866, 458601, 699556, 1048476, 1546116, 2246244, 3218644, 4553484, 6365684, 8801104, 12042732, 16319252, 21913612, 29174652, 38528732, 50495236, 65702076, 84906041

Offset: 0

Wolfdieter Lang, Apr 06 2001

Partial sums of A038165.

Colin Barker, Table of n, a(n) for n = 0..1000
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 20.
Index entries for linear recurrences with constant coefficients, signature (6,-10,-10,50,-34,-66,110,0,-110,66,34,-50,10,10,-6,1).

Accumulate[CoefficientList[Series[1/((1-x)(1-x^2))^5,{x,0,35}],x]] (* or *) LinearRecurrence[ {6,-10,-10,50,-34,-66,110,0,-110,66,34,-50,10,10,-6,1},{1,6,26,86,246,622,1442,3102,6292,12122,22374,39754,68354,114114,185614,294866},30] (* Harvey P. Dale, Mar 06 2016 *)

Vec(1/ ((1-x)^11*(1+x)^5) + O(x^40)) \\ Colin Barker, Jan 17 2017

a(n)= sum(A060098(n+5, 5)).
G.f.: 1/((1-x^2)^5*(1-x)^6) = 1/((1-x)^11*(1+x)^5).
a(n) = (14175*(30827+1941*(-1)^n) + 1440*(676427+11445*(-1)^n)*n + 126*(6861329+27375*(-1)^n)*n^2 + 1600*(258451+189*(-1)^n)*n^3 + 10*(12016607+945*(-1)^n)*n^4 + 22444800*n^5 + 2754192*n^6 + 220800*n^7 + 11130*n^8 + 320*n^9 + 4*n^10)/ 464486400. - Colin Barker, Jan 17 2017

A059601 Expansion of (1+10x+5x^2)/(1-x)^10.

Original entry on oeis.org

1, 20, 160, 820, 3190, 10252, 28600, 71500, 163735, 348920, 700128, 1334840, 2435420, 4276520, 7261040, 11966504, 19203965, 30091820, 46147200, 69397900, 102518130, 148991700, 213306600, 301185300

Offset: 0

Wolfdieter Lang, Feb 02 2001

a(n)= A038165(2*n).

Index entries for linear recurrences with constant coefficients, signature (10, -45, 120, -210, 252, -210, 120, -45, 10, -1).

a(n)= binomial(n+7, 7)*(4*n^2+23*n+18)/18.

G.f.:(1+10*x+5*x^2)/(1-x)^10.

A059602 Expansion of (5+10*x+x^2)/(1-x)^10.

Original entry on oeis.org

5, 60, 376, 1660, 5830, 17380, 45760, 109252, 240955, 497640, 972400, 1812200, 3241628, 5594360, 9354080, 15206840, 24107105, 37360004, 56722600, 84527300, 123830850, 178592700, 253886880, 356151900

Offset: 0

Wolfdieter Lang, Feb 02 2001

a(n)= A038165(2*n+1).

CoefficientList[Series[(5+10x+x^2)/(1-x)^10,{x,0,50}],x]  (* Harvey P. Dale, Mar 01 2011 *)

a(n)= binomial(n+7, 7)*(4*n^2+41*n+90)/18.

G.f.: (5+10*x+x^2)/(1-x)^10.

cp's OEIS Frontend

A059594 Convolution triangle based on A008619 (positive integers repeated).

Views

Author

Keywords

Comments

Examples

Programs

Mathematica

Maxima

Formula

A038166 G.f.: 1/((1-x)*(1-x^2))^6.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A060101 Sixth column (m=5) of triangle A060098.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

PARI

Formula

A059601 Expansion of (1+10*x+5*x^2)/(1-x)^10.

Views

Author

Keywords

Comments

Links

Formula

A059602 Expansion of (5+10*x+x^2)/(1-x)^10.

Views

Author

Keywords

Comments

Programs

Mathematica

Formula

A059601 Expansion of (1+10x+5x^2)/(1-x)^10.