A059718 -id:A059718 - cp's OEIS frontend

A059720 Triangle T(n,k), 0<=k<=n, formed from coefficients when formula for n-th diagonal of triangle in A059718 is written as a sum of binomial coefficients.

1, 0, 1, 0, 2, 1, 0, 5, 6, 2, 0, 15, 29, 20, 5, 0, 55, 148, 158, 80, 16, 0, 239, 818, 1185, 910, 366, 61, 0, 1199, 4964, 9094, 9392, 5696, 1904, 272, 0, 6810, 32989, 73026, 94833, 77011, 38719, 11080, 1385, 0, 43108, 238931, 619904, 970152, 988040, 663904, 285424, 71424, 7936

Offset: 0

N. J. A. Sloane, Feb 09 2001

I would very much like to find a formula for this - N. J. A. Sloane.

			1; 0,1; 0,2,1; 0,5,6,2; 0,15,29,20,5; ... E.g. the n=3 diagonal in A059718 has the formula b(m) = 0 + 5*m + 6*C(m,2) + 2*C(m,3) and so the third row here is 0, 5, 6, 2.

Interesting because it connects a mysterious sequence (A059219, the left edge) with a known sequence (A000111, the right edge). Cf. A059724, A059725, A059726.

A059216 Variation of Boustrophedon transform applied to all-1's sequence (see Comments for details).

Original entry on oeis.org

1, 2, 5, 14, 45, 169, 740, 3721, 21142, 133850, 933770, 7114115, 58758459, 522892624, 4987285553, 50751731950, 548839590949, 6285265061237, 75985249771496, 967047685739501, 12923640789599709, 180945893711983990, 2648725169100050894

Offset: 1

Floor van Lamoen, Jan 18 2001

Variation of Boustrophedon transform applied to all-1's sequence. Fill an array by diagonals in alternating directions - 'up' and 'down'. The first element of each diagonal is 1. When 'going up', add to the previous element the elements of the row the new element is in. When 'going down', add to the previous element the elements of the column the new element is in. The final element of the n-th diagonal is a(n).

			The array begins
   1  2  1 14  1 ...
   1  3 10 15 ...
   5  6 26 ...
   1 37 ...
  45 ...

G. C. Greubel, Table of n, a(n) for n = 1..480
Index entries for sequences related to boustrophedon transform

Cf. A000667, A059217, A059219, A059220, A059718.

# To get the array used to produce this sequence:
aaa := proc(m,n) option remember; local i,j,r,s,t1; if m=0 and n=0 then RETURN(1); fi; if n = 0 and m mod 2 = 1 then RETURN(1); fi; if m = 0 and n mod 2 = 0 then RETURN(1); fi; s := m+n; if s mod 2 = 1 then t1 := aaa(m+1,n-1); for j from 0 to n-1 do t1 := t1+aaa(m,j); od: else t1 := aaa(m-1,n+1); for j from 0 to m-1 do t1 := t1+aaa(j,n); od: fi; RETURN(t1); end; # the n-th antidiagonal in the up direction is aaa(n,0), aaa(n-1,1), aaa(n-2,2), ..., aaa(0,n)
# To get the array formed when the transformation is applied to an arbitrary input sequence b = [b[1], b[2], ..., b[N]]:
aab := proc(b,N,m,n) local i, j, r, s, t1; option remember; if m>N or n>N then error "asking for too many terms"; fi; if m = 0 and n mod 2 = 0 then RETURN(b[n+1]) end if; if n = 0 and m mod 2 = 1 then RETURN(b[m+1]) end if; s := m + n; if s mod 2 = 1 then t1 := aab(b,N,m + 1, n - 1); for j from 0 to n - 1 do t1 := t1 + aab(b,N,m, j) end do else t1 := aab(b,N,m - 1, n + 1); for j from 0 to m - 1 do t1 := t1 + aab(b,N,j, n) end do end if; RETURN(t1) end proc;
# To get the output sequence when the transformation is applied to an arbitrary input sequence b = [b[1], b[2], ..., b[N]]:
ff := proc(b) local N,t1,i; N := min(35, nops(b)); t1 := []; for i from 0 to N-1 do if i mod 2 = 0 then t1 := [op(t1),aab(b,N,i,0)]; else t1 := [op(t1),aab(b,N,0,i)]; fi; od: t1; end;

max = 22; t[0, 0] = 1; t[0, ?EvenQ] = 1; t[?OddQ, 0] = 1; t[n_, k_] /; OddQ[n + k](*up*):= t[n, k] = t[n + 1, k - 1] + Sum[t[n, j], {j, 0, k - 1}]; t[n_, k_] /; EvenQ[n + k](*down*):= t[n, k] = t[n - 1, k + 1] + Sum[t[j, k], {j, 0, n - 1}]; tnk = Table[t[n, k], {n, 0, max}, {k, 0, max - n}]; Join[{1}, Rest[Union[tnk[[1]], tnk[[All, 1]]]]] (* Jean-François Alcover, Jun 15 2012 *)

More terms from N. J. A. Sloane and Larry Reeves (larryr(AT)acm.org), Jan 23 2001

A059219 Variation of Boustrophedon transform applied to sequence 1,0,0,0,...: fill an array by diagonals in alternating directions - 'up' and 'down'. The first element of each diagonal after the first is 0. When 'going up', add to the previous element the elements of the row the new element is in. When 'going down', add to the previous element the elements of the column the new element is in. The final element of the n-th diagonal is a(n).

Original entry on oeis.org

1, 1, 2, 5, 15, 55, 239, 1199, 6810, 43108, 300731, 2291162, 18923688, 168402163, 1606199354, 16345042652, 176758631046, 2024225038882, 24471719797265, 311446235344127, 4162172487402027, 58275220793611957, 853045299274146032

Offset: 0

N. J. A. Sloane, Jan 18 2001

			The array begins
1 1 0 5 0 55 0 ...
0 1 3 5 48 55 ...
2 2 8 39 103 ...
0 12 27 152 ...
15 15 190 ...
0 221 ...

Vincenzo Librandi, Table of n, a(n) for n = 0..120
Index entries for sequences related to boustrophedon transform

Cf. A000667, A059216, A059217, A059220, A059237, A059720, A059718.

aaa := proc(m,n) option remember; local j,s,t1; if m=0 and n=0 then RETURN(1); fi; if n = 0 and m mod 2 = 1 then RETURN(0); fi; if m = 0 and n mod 2 = 0 then RETURN(0); fi; s := m+n; if s mod 2 = 1 then t1 := aaa(m+1,n-1); for j from 0 to n-1 do t1 := t1+aaa(m,j); od: else t1 := aaa(m-1,n+1); for j from 0 to m-1 do t1 := t1+aaa(j,n); od: fi; RETURN(t1); end; # the n-th antidiagonal in the up direction is aaa(n,0), aaa(n-1,1), aaa(n-2,2), ..., aaa(0,n)

max = 22; t[0, 0] = 1; t[0, ?EvenQ] = 0; t[?OddQ, 0] = 0; t[n_, k_] /; OddQ[n+k](* up *):= t[n, k] = t[n+1, k-1] + Sum[t[n, j], {j, 0, k-1}]; t[n_, k_] /; EvenQ[n+k](* down *):= t[n, k] = t[n-1, k+1] + Sum[t[j, k], {j, 0, n-1}]; tnk = Table[t[n, k], {n, 0, max}, {k, 0, max-n}]; Join[{1},  Rest[Union[tnk[[1]], tnk[[All, 1]]]]](* Jean-François Alcover, May 16 2012 *)

More terms from Floor van Lamoen, Jan 19 2001; and from N. J. A. Sloane Jan 20 2001.

A076038 Square array read by ascending antidiagonals in which row n has g.f. C/(1-nxC) where C = (1/2-1/2(1-4x)^(1/2))/x = g.f. for Catalan numbers A000108.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 5, 5, 1, 4, 10, 14, 14, 1, 5, 17, 35, 42, 42, 1, 6, 26, 74, 126, 132, 132, 1, 7, 37, 137, 326, 462, 429, 429, 1, 8, 50, 230, 726, 1446, 1716, 1430, 1430, 1, 9, 65, 359, 1434, 3858, 6441, 6435, 4862, 4862, 1, 10, 82, 530, 2582, 8952, 20532, 28770, 24310, 16796, 16796

Offset: 0

N. J. A. Sloane, Oct 29 2002

			Array begins as:
  1 1  2  5  14  42 ... (n=0)
  1 2  5 14  42 132 ... (n=1)
  1 3 10 35 126 ... (n=2)
  1 4 17 74 326 ...
  ...

Rows give A000108, A000108, A001700, A049027, A076025, A076026. Cf. A076037, A067347.

Cf. A059718.

Unprotect[Power]; Power[0,0]=1; Protect[Power]; A[n_, m_]:= 1/(m+1)*Sum[Binomial[2*m-k, m]*(k+1)*(n-m)^k,{k,0,m}]; Table[A[n,m],{n,0,10},{m,0,n}]//Flatten (* Stefano Spezia, Sep 01 2025 *)

A(n, m) = 1/(m+1)*Sum_{k=0..m} binomial(2*m-k, m)*(k+1)*(n-m)^k, m=0..n.

More terms from Vladeta Jovovic, Jul 18 2003

a(63)-a(65) from Stefano Spezia, Sep 01 2025

A063415 Triangle of coefficients of di-Boustrophedon transform (see A063179) read by rows: Let the original sequence be (U0,U1,...) and the transformed sequence (V0,V2,...), then Vn is a linear combination of U0,...,Un. T(n,m) is the coefficient that goes with Um to get Vn.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 12, 14, 9, 4, 1, 42, 51, 32, 14, 5, 1, 178, 214, 137, 60, 20, 6, 1, 870, 1049, 668, 295, 100, 27, 7, 1, 4830, 5820, 3713, 1636, 555, 154, 35, 8, 1, 29976, 36125, 23036, 10160, 3446, 952, 224, 44, 9, 1, 205572, 247734, 157993, 69664

Offset: 0

Floor van Lamoen, Jul 19 2001

			The triangle begins:
......1
....1...1
..2...2...1
4...5...3...1

T(n, 0) is A063179. Row sums form A062704. T(n, n-2) is A000096. Cf. A059718.

cp's OEIS Frontend

A059720 Triangle T(n,k), 0<=k<=n, formed from coefficients when formula for n-th diagonal of triangle in A059718 is written as a sum of binomial coefficients.

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A059216 Variation of Boustrophedon transform applied to all-1's sequence (see Comments for details).

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A076038 Square array read by ascending antidiagonals in which row n has g.f. C/(1-n*x*C) where C = (1/2-1/2*(1-4*x)^(1/2))/x = g.f. for Catalan numbers A000108.

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A063415 Triangle of coefficients of di-Boustrophedon transform (see A063179) read by rows: Let the original sequence be (U0,U1,...) and the transformed sequence (V0,V2,...), then Vn is a linear combination of U0,...,Un. T(n,m) is the coefficient that goes with Um to get Vn.

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Examples

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A076038 Square array read by ascending antidiagonals in which row n has g.f. C/(1-nxC) where C = (1/2-1/2(1-4x)^(1/2))/x = g.f. for Catalan numbers A000108.