A249388 -id:A249388 - cp's OEIS frontend

A245618 Triangle {H(n,k)} similar to Pascal's with sides of 1's, but interior entries are obtained by the rule: H(n,k) = |H(n-1,k)+(-1)^m(n,k)*H(n-1,k-1)|, where m(n,k) = H(n-1,k) + H(n-1,k-1).

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 4, 4, 1, 1, 1, 2, 3, 8, 3, 2, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 2, 2, 6, 10, 6, 2, 2, 1, 1, 1, 4, 8, 16, 16, 8, 4, 1, 1, 1, 2, 3, 12, 24, 32, 24, 12, 3, 2, 1, 1, 1, 1, 9, 36, 56, 56, 36, 9, 1, 1, 1, 1, 2, 2, 10

Offset: 0

Vladimir Shevelev, Nov 05 2014

Let us consider the operation <+> over integers such that k<+>m = |k+(-1)^(k+m)*m|. Then H(n,k) = H(n-1,k)<+>H(n-1,k-1).

This is an analog of the formula binomial(n,k) = binomial(n-1,k) + binomial(n-1,k-1).

			Triangle begins
1
1  1
1  2  1
1  1  1  1
1  2  2  2  1
1  1  4  4  1  1
1  2  3  8  3  2  1
....................

Peter J. C. Moses, First 50 rows.

Cf. A007318, row sums in A245619, row "sums", using <+>, in A249388.

parityAdd[a_, b_] := Abs[a + b (-1)^(a + b)];
triangleHP[n_, 0] := 1;
triangleHP[n_, n_] := 1;
triangleHP[n_, k_] := triangleHP[n, k] = parityAdd[triangleHP[n - 1, k - 1], triangleHP[n - 1, k]];
Flatten[Table[triangleHP[n, k], {n, 0, 15}, {k, 0, n}]] (* Peter J. C. Moses, Nov 05 2014 *)

More terms from Peter J. C. Moses, Nov 05 2014

A249779 Row "sums" of Pascal triangle (A007318), using operation <+> defined in comment in A245618.

Original entry on oeis.org

1, 2, 2, 2, 2, 22, 20, 28, 2, 494, 912, 1672, 2376, 4836, 4160, 4184, 2, 131038, 261800, 522272, 1035804, 2053288, 3977272, 7742352, 13942968, 28016020, 47111040, 84948528, 92072064, 272727022, 249686810, 167376688, 2, 8589934526, 17179867992, 34359725136

Offset: 0

Vladimir Shevelev, Nov 05 2014

nonn

Operation <+> is defined in A245618 as: k<+>m = |k+(-1)^(k+m)*m|.

a(n)=2 for n=1,2,3,4,8,16,32,64,128,256,...

			For n=4, we have row 1,4,6,4,1.
By definition of <+>, we find 1<+>4=3, 3<+>6=3, 3<+>4=1, 1<+>1=2. So a(4)=2.

Michael De Vlieger, Table of n, a(n) for n = 0..1000

Cf. A007318, A245618, A245619, A249388.

a249779[n_Integer] := Module[{m0082, pls, lst},
  m0082[j_] := Table[Binomial[j, k], {k, 0, j}];
  pls[k_, m_] := Abs[k + (-1)^(k + m)*m];
  lst = m0082[n];
  For[i = 0, i < n, i++, lst[[2]] = pls[lst[[1]], lst[[2]]];
   lst = Drop[lst, 1]];
  lst[[1]]
]; a249779 /@ Range[35] (* Michael De Vlieger, Nov 23 2014 *)
parityAdd[a_,b_]:=Abs[a+b (-1)^(a+b)];
Map[Fold[parityAdd,First[#],Rest[#]]&[Binomial[#,Range[0,#]]]&,Range[0,35]] (* Peter J. C. Moses, Dec 01 2014 *)

More terms from Peter J. C. Moses, Nov 05 2014

A249416 a(n) = core(Sum_{i=0,...,n} core(binomial(n,i))), where core(n) = A007913(n).

Original entry on oeis.org

1, 2, 1, 2, 10, 2, 1, 2, 118, 19, 519, 2, 635, 370, 829, 1333, 8454, 17315, 3599, 15307, 423769, 852006, 495431, 2, 2425755, 2121070, 3192295, 1614598, 35685686, 10081687, 735961, 12902173, 216093318, 151123623, 5270424935, 39937013, 22884337, 7281379334

Offset: 0

Vladimir Shevelev, Oct 28 2014

nonn

Cf. A007913, A248470, A248473, A249388.

a7913[n_]:=a7913[n]=Times@@(#[[1]]^Mod[#[[2]],2])&[Transpose[FactorInteger[n]]];
Map[a7913[Total[Map[a7913,Binomial[#,Range[0,#]]]]]&,Range[0,50]] (* Peter J. C. Moses, Oct 28 2014 *)

a(n) = core(sum(i=0, n, core(binomial(n,i)))); \\ Michel Marcus, Nov 13 2014

More terms from Peter J. C. Moses, Oct 28 2014

cp's OEIS Frontend

A245618 Triangle {H(n,k)} similar to Pascal's with sides of 1's, but interior entries are obtained by the rule: H(n,k) = |H(n-1,k)+(-1)^m(n,k)*H(n-1,k-1)|, where m(n,k) = H(n-1,k) + H(n-1,k-1).

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A249779 Row "sums" of Pascal triangle (A007318), using operation <+> defined in comment in A245618.

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A249416 a(n) = core(Sum_{i=0,...,n} core(binomial(n,i))), where core(n) = A007913(n).

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