A090447 -id:A090447 - cp's OEIS frontend

A090448 Fourth column (m=3) of triangle A090447.

9, 96, 500, 1800, 5145, 12544, 27216, 54000, 99825, 174240, 290004, 463736, 716625, 1075200, 1572160, 2247264, 3148281, 4332000, 5865300, 7826280, 10305449, 13406976, 17250000, 21970000, 27720225, 34673184, 43022196, 52983000, 64795425, 78725120, 95065344

Offset: 3

Wolfdieter Lang, Dec 23 2003

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000292, A000330, A090447.

seq(mul(binomial(n,k),k=1..3),n=3..30); # Zerinvary Lajos, Dec 13 2007

a[n_] := Product[Binomial[n, k], {k, 0, 3}]; Array[a, 30, 3] (* Amiram Eldar, Sep 08 2022 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{9,96,500,1800,5145,12544,27216},40] (* Harvey P. Dale, Jul 18 2025 *)

a(n) = A090447(n,3).
a(n) = (n^3*(n-1)^2*(n-2)^1)/(1!*2!*3!) for n >= 3.
From Colin Barker, Jan 21 2013: (Start)
a(n) = (n^6-4*n^5+5*n^4-2*n^3)/12.
G.f.: -x^3*(x^3+17*x^2+33*x+9)/(x-1)^7. (End)
a(n) = A000330(n-1)^2 - A000292(n-1)^2. - J. M. Bergot, May 02 2014
a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Wesley Ivan Hurt, May 04 2021
From Amiram Eldar, Sep 08 2022: (Start)
Sum_{n>=3} 1/a(n) = 207/4 - 9*Pi^2/2 - 6*zeta(3).
Sum_{n>=3} (-1)^(n+1)/a(n) = 165/4 - Pi^2/4 - 48*log(2) - 9*zeta(3)/2. (End)

A090449 Fifth column (m=4) of triangle A090447.

Original entry on oeis.org

96, 2500, 27000, 180075, 878080, 3429216, 11340000, 32942250, 86248800, 207352860, 464199736, 978193125, 1956864000, 3741740800, 6876627840, 12202737156, 20988540000, 35103820500, 57249238200, 91254750895, 142462526976, 218212500000, 328451500000, 486489948750

Offset: 4

Wolfdieter Lang, Dec 23 2003

T. D. Noe, Table of n, a(n) for n = 4..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A090447.

LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{96,2500,27000,180075,878080,3429216,11340000,32942250,86248800,207352860,464199736},40] (* Harvey P. Dale, Apr 10 2018 *)

a(n)= A090447(n, 4)= (n^4*(n-1)^3*(n-2)^2*(n-3)^1)/(1!*2!*3!*4!), n>=4.
G.f.: -x^4*(x^6+109*x^5+1435*x^4+4735*x^3+4780*x^2+1444*x+96)/(x-1)^11. - Colin Barker, Jan 21 2013
From Amiram Eldar, Sep 08 2022: (Start)
Sum_{n>=4} 1/a(n) = 700*Pi^2/9 + 4*Pi^4/15 - 40*zeta(3) - 20129/27.
Sum_{n>=4} (-1)^n/a(n) = 30311/27 - 26*Pi^2/9 - 7*Pi^4/30 - 11008*log(2)/9 - 186*zeta(3). (End)

A090450 Row sums of triangle A090447.

Original entry on oeis.org

1, 2, 5, 22, 221, 5556, 352897, 56909000, 23456163817, 24862388706622, 68125408037874461, 484697617517152241192, 8987761653844408528516645, 435751632681268773059903563592, 55389930392445599773550918556967161

Offset: 0

Wolfdieter Lang, Dec 23 2003

Cf. A090451 (alternating row sums).

a(n)=sum(A090447(n, m), m=0..n), n>=0.

A090451 Alternating row sums of triangle A090447.

Original entry on oeis.org

1, 0, 1, -2, 21, -454, 25285, -3606504, 1328522713, -1270747453940, 3169192850406441, -20676520009454537480, 353872602524737587995341, -15925173912641846013871947522, 1888348697181821236021540428910349, -591053910458037676348857289166882470064

Offset: 0

Wolfdieter Lang, Dec 23 2003

a(n)=sum(A090447(n, m)*(-1)^m, m=0..n), n>=0.

A090450 (row sums).

A120409 a(n) = n^1(n+1)^2(n+2)^3(n+3)^4(n+4)^5(n+5)^6/(1!2!3!4!5!6!).

Original entry on oeis.org

162000, 26471025, 1376829440, 36294822144, 600112800000, 7031325609000, 63117561830400, 457937132487120, 2790771598030416, 14702257341646875, 68449036271616000, 286552568263270400, 1093771338292039680, 3849852478998931776, 12612749124441600000

Offset: 1

Zerinvary Lajos, Jul 05 2006

Index entries for linear recurrences with constant coefficients, signature (22,-231,1540,-7315,26334,-74613,170544,-319770,497420,-646646, 705432,-646646,497420,-319770,170544,-74613,26334,-7315,1540,-231,22,-1).

Cf. A090447, A090448, A090449.

[seq(n^1*(n+1)^2*(n+2)^3*(n+3)^4*(n+4)^5*(n+5)^6/(1!*2!*3!*4!*5!*6!),n=1..27)];

Table[(Times@@Table[(n+k)^(k+1),{k,0,5}])/Times@@(Range[6]!),{n,15}] (* Harvey P. Dale, Jun 07 2022 *)

[binomial(n,1)*binomial(n,3)*binomial(n,5)*binomial(n,2)*binomial(n,4)*binomial(n,6) for n in range(6, 19)] # Zerinvary Lajos, May 17 2009

Sum_{n>=1} 1/a(n) = 789878089*Pi^2/18000 + 64687*Pi^4/150 - 16*Pi^6/21 + 6603436*zeta(3)/25 + 80136*zeta(5) - 56698539425671/64800000. - Amiram Eldar, Sep 08 2022

Offset changed from 0 to 1 by Georg Fischer, May 08 2021

A120408 a(n) = n^1(n+1)^2(n+2)^3(n+3)^4(n+4)^5/(1!2!3!4!5!).

Original entry on oeis.org

2500, 162000, 3781575, 49172480, 432081216, 2857680000, 15219319500, 68309049600, 266863130820, 929327871472, 2937513954375, 8547581952000, 23153892070400, 58918947333120, 141893427649968, 325406324160000, 714327643354500, 1507601438758800, 3070631112865855

Offset: 1

Zerinvary Lajos, Jul 05 2006

Cf. A090447, A090448, A090449, A120409.

[seq(n^1*(n+1)^2*(n+2)^3*(n+3)^4*(n+4)^5/(1!*2!*3!*4!*5!),n=1..37)];

Table[n^1*(n+1)^2*(n+2)^3*(n+3)^4*(n+4)^5/(1!*2!*3!*4!*5!),{n,19}] (* James C. McMahon, Oct 05 2024 *)

Offset changed from 0 to 1 by Georg Fischer, May 08 2021

a(17)-a(19) from James C. McMahon, Oct 05 2024

A120410 a(n) = n^1(n+1)^2(n+2)^3(n+3)^4(n+4)^5(n+5)^6(n+6)^7/(1!2!3!4!5!6!7!).

Original entry on oeis.org

0, 26471025, 11014635520, 1306613597184, 72013536000000, 2320337450970000, 49989108969676800, 785820119347897920, 9577928124440387712, 94609025993497640625, 783056974947287040000, 5572874347584082739200, 34808179069805870776320, 193986366711798174329088

Offset: 0

Zerinvary Lajos, Jul 05 2006

Index entries for linear recurrences with constant coefficients, signature (29,-406,3654,-23751,118755,-475020,1560780,-4292145,10015005,-20030010,34597290,-51895935,67863915,-77558760, 77558760,-67863915,51895935,-34597290,20030010,-10015005,4292145,-1560780,475020,-118755,23751,-3654,406,-29,1).

Cf. A090447 A090448 A090449.

[seq(n^1*(n+1)^2*(n+2)^3*(n+3)^4*(n+4)^5*(n+5)^6*(n+6)^7/(1!*2!*3!*4!*5!*6!*7!),n=1..17)];

Table[n*(n+1)^2*(n+2)^3*(n+3)^4*(n+4)^5*(n+5)^6*(n+6)^7/(1!*2!*3!*4!*5!*6!*7!), {n, 0, 10}] (* Amiram Eldar, Sep 08 2022 *)

Sum_{n>=1} 1/a(n) = 422971791896349857/972000000 - 845737633741*Pi^2/22500 - 230834541*Pi^4/500 - 58492*Pi^6/15 - 18320341039*zeta(3)/1800 - 15501934*zeta(5)/5 - 5040*zeta(7). - Amiram Eldar, Sep 08 2022

a(0) prepended by Amiram Eldar, Sep 08 2022

cp's OEIS Frontend

A090448 Fourth column (m=3) of triangle A090447.

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A090449 Fifth column (m=4) of triangle A090447.

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A090450 Row sums of triangle A090447.

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A090451 Alternating row sums of triangle A090447.

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A120409 a(n) = n^1*(n+1)^2*(n+2)^3*(n+3)^4*(n+4)^5*(n+5)^6/(1!*2!*3!*4!*5!*6!).

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A120408 a(n) = n^1*(n+1)^2*(n+2)^3*(n+3)^4*(n+4)^5/(1!*2!*3!*4!*5!).

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A120410 a(n) = n^1*(n+1)^2*(n+2)^3*(n+3)^4*(n+4)^5*(n+5)^6*(n+6)^7/(1!*2!*3!*4!*5!*6!*7!).

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A120409 a(n) = n^1(n+1)^2(n+2)^3(n+3)^4(n+4)^5(n+5)^6/(1!2!3!4!5!6!).

A120408 a(n) = n^1(n+1)^2(n+2)^3(n+3)^4(n+4)^5/(1!2!3!4!5!).

A120410 a(n) = n^1(n+1)^2(n+2)^3(n+3)^4(n+4)^5(n+5)^6(n+6)^7/(1!2!3!4!5!6!7!).