A112357 -id:A112357 - cp's OEIS frontend

A112356 Following triangle is based on Pascal's triangle. The r-th term of the n-th row is product of C(n,r) successive integers such that the product of all the terms of the row is (2^n)!. Sequence contains the triangle read by rows.

1, 1, 2, 1, 6, 4, 1, 24, 210, 8, 1, 120, 332640, 32760, 16, 1, 720, 29059430400, 19275223968000, 20389320, 32, 1, 5040, 223016017416192000, 1250004633476421848894668800000, 28844656968251942737920000, 48920775120, 64

Offset: 0

Amarnath Murthy, Sep 05 2005

The leading diagonal contains 2^n. The second column terms are (n+1)!.

			Triangle begins:
  1
  1 2
  1 6 4
  1 24 210 8
  1 120 332640 32760 16
...
The row for n = 3 is
  1 3 3 1
  1 (2*3*4) (5*6*7) 8 or (1 24 210 8)

Cf. A112357.

A112356(n)= { local(resul,piv,a); resul=[1]; piv=2; for(col=1,n, a=piv; piv++; for(c=2,binomial(n,col), a *= piv; piv++; ); resul=concat(resul,a); ); return(resul); }
{ for(row=0,7, print(A112356(row)); ); } \\ R. J. Mathar, May 19 2006

More terms from Mandy Stoner (astoner(AT)ashland.edu), Apr 27 2006

A112358 The following triangle is based on Pascal's triangle. The r-th term of the n-th row is sum of C(n,r) successive integers so that the sum of all the terms of the row is (2^n)*(2^n+1)/2, the 2^n -th triangular number. Sequence contains the triangle read by rows.

Original entry on oeis.org

1, 1, 2, 1, 5, 4, 1, 9, 18, 8, 1, 14, 51, 54, 16, 1, 20, 115, 215, 145, 32, 1, 27, 225, 650, 750, 363, 64, 1, 35, 399, 1645, 2870, 2310, 868, 128, 1, 44, 658, 3668, 8995, 10724, 6538, 2012, 256, 1, 54, 1026, 7434, 24381, 40257, 35658, 17442, 4563, 512, 1, 65, 1530, 13980, 59115, 129150, 156135, 109020, 44595, 10185, 1024

Offset: 0

Amarnath Murthy, Sep 05 2005

The leading diagonal contains 2^n.

			Row for n = 3 is 1, (2+3+4), (5+6+7), 8.
Triangle begins:
  1
  1 2
  1 5 4
  1 9 18 8
  1 14 51 54 16
  ...

Cf. A112356, A112357, A112359.

Cf. A008949.

A008949[n_, k_] := Sum[Binomial[n, j], {j, 0, k}];
A112358[n_, k_] := If[k == 0, 1, Binomial[A008949[n, k] + 1, 2] - Binomial[A008949[n, k - 1] + 1, 2]];
Table[A112358[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 01 2023 *)

T(n,0) = 1, T(n,k) = C(A008949(n,k)+1, 2) - C(A008949(n,k-1)+1, 2) = C(n,k)*(A008949(n+1,k)+1)/2 for k>0. - Franklin T. Adams-Watters, Sep 27 2006

More terms from Amber Reardon (alr5041(AT)psu.edu) and Vincent M. DelPrince (vmd5003(AT)psu.edu), Oct 04 2005

A112359 Product of n-th row of A112358.

Original entry on oeis.org

1, 2, 20, 1296, 616896, 2294480000, 68803020000000, 16921170978243840000, 34496793424028349312532480, 587395062985562798532990766497792, 84034508984208959408391703340160000000000, 101510206136861741998326287566434701976960000000000

Offset: 0

Amarnath Murthy, Sep 05 2005

Cf. A112356, A112357, A112358.

A008949 := proc(n,k) local i ; add(binomial(n,i),i=0..k) ; end: A112358 := proc(n,k) if k = 0 then 1 ; else binomial(n,k)*(A008949(n+1,k)+1)/2 ; fi ; end: A112359 := proc(n) local k ; mul( A112358(n,k),k=0..n ) ; end: for n from 0 to 15 do printf("%d, ",A112359(n)) ; od ; # R. J. Mathar, May 08 2007

A008949[n_, k_] := Sum[Binomial[n, j], {j, 0, k}];
A112358[n_, k_] := If[k == 0, 1, Binomial[A008949[n, k] + 1, 2] - Binomial[A008949[n, k - 1] + 1, 2]];
a[n_] := Product[A112358[n, k], {k, 0, n}];
Table[a[n], {n, 0, 11}] (* Jean-François Alcover, Aug 01 2023, after R. J. Mathar *)

More terms from R. J. Mathar, May 08 2007

A112360 Triangle read by rows: T(n,k) is the LCM of all C(n,k) integers from 1 + C(n,0) + C(n,1) + ... + C(n,k-1) to C(n,0) + C(n,1) + ... + C(n,k) (0 <= k <= n).

Original entry on oeis.org

1, 1, 2, 1, 6, 4, 1, 12, 210, 8, 1, 60, 27720, 5460, 16, 1, 60, 720720, 13385572200, 3398220, 32, 1, 420, 232792560, 219060189739591200, 60218289392461200, 4076731260, 64, 1, 840, 2329089562800, 1182266884102822267511361600

Offset: 0

Amarnath Murthy, Sep 07 2005

Column 1 yields A003418. a(n,n) = 2^n. - Emeric Deutsch, Feb 03 2006

			Triangle begins:
  1;
  1,  2;
  1,  6,     4;
  1, 12,   210,    8;
  1, 60, 27720, 5460, 16;
  ...
The row for n = 3 is
1....3..........3.......1
1 lcm(2*3*4) lcm(5*6*7) 8 ====> 1 12 210 8.

Cf. A112356, A112357, A112358, A112359.

T:=proc(n,k) if n=0 and k=0 then 1 else lcm(seq(j,j=1+sum(binomial(n,i),i=0..k-1)..sum(binomial(n,i),i=0..k))) fi end: for n from 0 to 7 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form - Emeric Deutsch, Feb 03 2006

More terms from Emeric Deutsch, Feb 03 2006

cp's OEIS Frontend

A112356 Following triangle is based on Pascal's triangle. The r-th term of the n-th row is product of C(n,r) successive integers such that the product of all the terms of the row is (2^n)!. Sequence contains the triangle read by rows.

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A112358 The following triangle is based on Pascal's triangle. The r-th term of the n-th row is sum of C(n,r) successive integers so that the sum of all the terms of the row is (2^n)*(2^n+1)/2, the 2^n -th triangular number. Sequence contains the triangle read by rows.

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A112359 Product of n-th row of A112358.

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A112360 Triangle read by rows: T(n,k) is the LCM of all C(n,k) integers from 1 + C(n,0) + C(n,1) + ... + C(n,k-1) to C(n,0) + C(n,1) + ... + C(n,k) (0 <= k <= n).

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