A090448 -id:A090448 - cp's OEIS frontend

A090447 Triangle of partial products of binomials.

1, 1, 1, 1, 2, 2, 1, 3, 9, 9, 1, 4, 24, 96, 96, 1, 5, 50, 500, 2500, 2500, 1, 6, 90, 1800, 27000, 162000, 162000, 1, 7, 147, 5145, 180075, 3781575, 26471025, 26471025, 1, 8, 224, 12544, 878080, 49172480, 1376829440, 11014635520, 11014635520, 1, 9, 324

Offset: 0

Wolfdieter Lang, Dec 23 2003

			[1]; [1,1]; [1,2,2]; [1,3,9,9]; ...

W. Lang, First 10 rows.

Column sequences: A000027 (natural numbers), A006002, A090448-9.
Cf. A090450 (row sums), A090451 (alternating row sums).
Cf. A008949 (partial row sums in Pascal's triangle).
Cf. A000178, A007318, A008279.

a[n_, m_] := Product[Binomial[n, p], {p, 0, m}]; Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Sep 01 2016 *)

a(n, m) = Product_{p=0..m} binomial(n, p), n>=m>=0, else 0. Partial row products in Pascal's triangle A007318.
a(n, m) = (Product_{p=0..m} fallfac(n, m-p))/superfac(m), n>=m>=0, else 0; with fallfac(n, m) := A008279(n, m) (falling factorials) and superfac(m) = A000178(m) (superfactorials).
a(n, m) = (Product_{p=0..m} (n-p)^(m-p))/superfac(m), n>=m>=0, with 0^0:=0, else 0.

A189346 Number of sets of four points on an n X n grid (or geoboard), exactly three of which are collinear.

Original entry on oeis.org

0, 0, 48, 532, 3088, 11340, 33824, 83288, 183344, 364304, 681872, 1194100, 1992976, 3182332, 4941360, 7420640, 10874720, 15539952, 21812720, 30011924, 40650368, 54187196, 71463440, 92990296, 119675712, 152314920, 192393872, 240690060

Offset: 1

Martin Renner, Apr 20 2011

nonn

The four points build a triangle on an n X n grid, with one of them located on a side of the triangle.

The number of sets of four points with the three collinear points in a horizontal or vertical line is 2*n^2*(n-1)*binomial(n,3) = 4*A090448(n). The number of sets of four points with the three collinear points in a diagonal line of slope 1 is 2*n*(n-1)*binomial(n,3) + 4*Sum_{k=3..n-1}(n^2-k)*binomial(k,3). The sum of these two values is a lower bound for this sequence. - Nathaniel Johnston, Apr 23 2011

Martin Renner, Table of n, a(n) for n = 1..76

Cf. A000938, A175383, A178256, A189345.

A189346 := proc(n)local a,b,j,k,l,m,s,slopes,num,den,tot: tot := 0: slopes := {}: for b from 1 to ceil(n/2)-1 do for a from 0 to b do slopes := slopes union {a/b}: od: od: for s from 1 to nops(slopes) do num := numer(slopes[s]): den := denom(slopes[s]): if(num = 0)then tot := tot + 2*n^2*(n-1)*binomial(n,3): elif(num = den)then tot := tot + 2*(2*add(binomial(k,3)*(n^2-k), k=3..n) - binomial(n,3)*(n^2 - n)): else for j from 1 to n - 2*den do for k from 1 to n - 2*num do tot := tot + 4*(n^2 - 3): for l from 1 to n do for m from 1 to n do if((not l = j or not m = k) and (not l = j + den or not m = k + num) and (not l = j + 2*den or not m = k + 2*num) and (m - k)*den = num*(l - j))then tot := tot - 4: fi: od: od: od: od: fi: od: return tot: end:
seq(A189346(n),n=1..15); # Nathaniel Johnston, Apr 23 2011

a(6)-a(28) from Nathaniel Johnston, Apr 23 2011

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

A090447 Triangle of partial products of binomials.

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A189346 Number of sets of four points on an n X n grid (or geoboard), exactly three of which are collinear.

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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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Sage

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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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Programs

Maple

Mathematica

Formula

Extensions

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!).