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Revision History for A156697 (Bold, blue-underlined text is an addition; faded, red-underlined text is a deletion.)

Showing entries 1-10 | older changes
Triangle T(n, k, m) = t(n,m)/( t(k,m) * t(n-k,m) ) with T(n, 0, m) = T(n, n, m) = 1, where t(n, m) = Product_{j=1..n} Product_{i=1..j-1} ( 1 - (m+1)*(2*i-1) ) and m = 2, read by rows.
(history; published version)
#11 by Michel Marcus at Mon Mar 01 02:01:44 EST 2021
STATUS

reviewed

approved

#10 by Joerg Arndt at Mon Mar 01 01:57:17 EST 2021
STATUS

proposed

reviewed

#9 by G. C. Greubel at Fri Feb 26 03:16:10 EST 2021
STATUS

editing

proposed

Discussion
Fri Feb 26
03:28
G. C. Greubel: In the first Mma program ( a cleaned version of the Bagula Mma) t(n, k) is used. seems ok in this sense since Mma doesn't really care. In terms of reading it still, to me, makes sense to use T(n,k,m) -> t(n,m)/(..) -> apply m=2.
04:33
Michel Marcus: you mean T(n, k) = t(n,2)/( t(k,2) * t(n-k,2) )  ??
13:17
G. C. Greubel: Yes, for this sequence it is of the form T(n,k) = t(n,2)/(t(k,2)*t(n-k,2)). In this Bagula set there are 12 sequences with nearly the same form (m=1,2,3; p, q see 2nd formula). My thinking is that keeping t(n,m) reduces the set to a fixed (n,k) and variable (m,p,q) and leads to the least amount of variable confusion (ie T(n,k,m,p,q) and t(n,k) with ex being m=4 means T(n,k,4,p,q) and t(n,4)).
#8 by G. C. Greubel at Fri Feb 26 03:10:27 EST 2021
NAME

Triangle T(n, k, m) = t(n,m)/( t(k,nm) * t(n-k,m) ) with T(n, 0, m) = T(n, n, m) = 1, where t(n, m) = Product_{j=1..n} Product_{i=1..j-1} ( 1 - (m+1)*(2*i-1) ) and m = 2, read by rows.

FORMULA

T(n, k, m) = t(n,m)/( t(k,nm) * t(n-k,m) ) with T(n, 0, m) = T(n, n, m) = 1, where t(n, m) = Product_{j=1..n} Product_{i=1..j-1} ( 1 - (m+1)*(2*i-1) ) and m = 2.

STATUS

reviewed

editing

Discussion
Fri Feb 26
03:16
G. C. Greubel: In a normal case t(n,k) would be the preferred representation. In this case chose, since T(n,k,m) ~ t(n, m), to keep the form of t(n, m) with m being fixed in the related sequences (m=1..3).
#7 by Joerg Arndt at Fri Feb 26 01:44:22 EST 2021
STATUS

proposed

reviewed

Discussion
Fri Feb 26
03:02
Michel Marcus: where t(n, m) = Product_{j=1..n} Product_{i=1..j-1} ( 1 - (m+1)*(2*i-1) ) should rather have k instead of m ? (because globally m=2 )
03:03
Michel Marcus: or I misunderstood ?
#6 by G. C. Greubel at Thu Feb 25 20:09:10 EST 2021
STATUS

editing

proposed

#5 by G. C. Greubel at Thu Feb 25 20:08:59 EST 2021
NAME

General qp odd weighted combinations as : m=2;q=3: t(n,m)=If[m == 0, n!, Product[Product[1 - (2*i - 1)*( m + 1), {i, 0, k - 1}], {k, 1, n}]]; b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])].

Triangle T(n, k, m) = t(n,m)/( t(k,n) * t(n-k,m) ) with T(n, 0, m) = T(n, n, m) = 1, where t(n, m) = Product_{j=1..n} Product_{i=1..j-1} ( 1 - (m+1)*(2*i-1) ) and m = 2, read by rows.

DATA

1, 1, 1, 1, -2, 1, 1, 16, 16, 1, 1, -224, 1792, -224, 1, 1, 4480, 501760, 501760, 4480, 1, 1, -116480, 260915200, -3652812800, 260915200, -116480, 1, 1, 3727360, 217081446400, 60782804992000, 60782804992000, 217081446400, 3727360, 1, 1

COMMENTS

Row sums are: {1, 2, 0, 34, 1346, 1012482, -3131215358, 121999780331522,

{1, 2, 0, 34, 1346, 1012482, -3131215358, 121999780331522,

34591292869081661442, 107137531255480378706493442, ...}.

-3432787564907030237721525583871998,...}.

LINKS

G. C. Greubel, <a href="/A156697/b156697.txt">Rows n = 0..30 of the triangle, flattened</a>

FORMULA

T(n, k, m) = t(n,m)/( t(k,n) * t(n-k,m) ) with T(n, 0, m) = T(n, n, m) = 1, where t(n, m) = Product_{j=1..n} Product_{i=1..j-1} ( 1 - (m+1)*(2;q*i-1) ) and m =3: 2.

tT(n, k, m, p, q) =If[ (-p*(m+1))^(k*(n-k)) * (f(n,m,p,q)/(f(k,m == 0, ,p,q)*f(n!, Product[-k,m,p,q))) where Product[_{j=1 - ..n} Pochhammer( (2q*i (m+1) - 1)/(p*( m + 1), {i, 0, k ), j) for (m, p, q) = (2, 2, - 1}], {k, 1, n}]];). - _G. C. Greubel_, Feb 25 2021

b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])].

EXAMPLE

{1},

Triangle begins as:

1;

{ 1, 1},;

{ 1, -2, 1},;

{ 1, 16, 16, 1},;

{ 1, -224, 1792, -224, 1},;

{ 1, 4480, 501760, 501760, 4480, 1},;

{ 1, -116480, 260915200, -3652812800, 260915200, -116480, 1},;

{1, 3727360, 217081446400, 60782804992000, 60782804992000, 217081446400, 3727360, 1},

{1, -141639680, 263971038822400, -1921709162627072000, 38434183252541440000, -1921709162627072000, 263971038822400, -141639680, 1},

{1, 6232145920, 441359576911052800, 102819127037198860288000, 53465946059343407349760000, 53465946059343407349760000, 102819127037198860288000, 441359576911052800, 6232145920, 1},

{1, -311607296000, 970991069204316160000, -8595679020309824720076800000, 143032098897955483342077952000000, -3718834571346842566894026752000000, 143032098897955483342077952000000, -8595679020309824720076800000, 970991069204316160000, -311607296000, 1}

MATHEMATICA

(* First program *)

t[n_, m_k_] := If[m k== 0, n!, Product[Product[1 - (2*i - 1)*( m k+ 1), {j, n}, {i, 0, k j- 1}], {k, 1, n}] ];

bT[n_, k_, m_] := If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];

Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 15}]

Table[T[n, k, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 25 2021 *)

(* Second program *)

f[n_, m_, p_, q_]:= Product[Pochhammer[(q*(m+1) -1)/(p*(m+1)), j], {j, n}];

T[n_, k_, m_, p_, q_]:= (-p*(m+1))^(k*(n-k))*(f[n, m, p, q]/(f[k, m, p, q]*f[n-k, m, p, q]));

Table[T[n, k, 2, 2, -1], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 25 2021 *)

PROG

(Sage)

@CachedFunction

def f(n, m, p, q): return product( rising_factorial( (q*(m+1)-1)/(p*(m+1)), j) for j in (1..n))

def T(n, k, m, p, q): return (-p*(m+1))^(k*(n-k))*(f(n, m, p, q)/(f(k, m, p, q)*f(n-k, m, p, q)))

flatten([[T(n, k, 2, 2, -1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 25 2021

(Magma)

f:= func< n, m, p, q | n eq 0 select 1 else m eq 0 select Factorial(n) else (&*[ 1 -(p*i+q)*(m+1): i in [0..j], j in [0..n-1]]) >;

T:= func< n, k, m, p, q | f(n, m, p, q)/(f(k, m, p, q)*f(n-k, m, p, q)) >;

[T(n, k, 2, 2, -1): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 25 2021

CROSSREFS

Cf. A007318 (m=0), A156696 (m=1), this sequence (m=2), A156698 (m=3).

Cf. A156690, A156691, A156692, A156693.

Cf. A156691, A156699, A156725.

KEYWORD

sign,tabl,uned

EXTENSIONS

Edited by G. C. Greubel, Feb 25 2021

STATUS

approved

editing

#4 by Charles R Greathouse IV at Wed May 15 18:39:22 EDT 2013
STATUS

editing

approved

#3 by Charles R Greathouse IV at Wed May 15 18:39:19 EDT 2013
MAPLE

t[n_, m_] = If[m == 0, n!, Product[Product[1 - (2*i - 1)*( m + 1), {i, 0, k - 1}], {k, 1, n}]];

b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];

Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 15}]

MATHEMATICA

t[n_, m_] = If[m == 0, n!, Product[Product[1 - (2*i - 1)*( m + 1), {i, 0, k - 1}], {k, 1, n}]];

b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];

Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 15}]

STATUS

approved

editing

#2 by Russ Cox at Fri Mar 30 17:34:33 EDT 2012
AUTHOR

_Roger L. Bagula (rlbagulatftn(AT)yahoo.com), _, Feb 13 2009

Discussion
Fri Mar 30
17:34
OEIS Server: https://oeis.org/edit/global/158