A034868 -id:A034868 - cp's OEIS frontend

A265848 Pascal's triangle, right and left halves interchanged.

1, 1, 1, 1, 1, 2, 3, 1, 1, 3, 4, 1, 1, 4, 6, 10, 5, 1, 1, 5, 10, 15, 6, 1, 1, 6, 15, 20, 35, 21, 7, 1, 1, 7, 21, 35, 56, 28, 8, 1, 1, 8, 28, 56, 70, 126, 84, 36, 9, 1, 1, 9, 36, 84, 126, 210, 120, 45, 10, 1, 1, 10, 45, 120, 210, 252, 462, 330, 165, 55, 11, 1, 1, 11, 55, 165, 330, 462

Offset: 0

Reinhard Zumkeller, Dec 24 2015

Concatenations of rows of A014413 and A034868.

Alternative mirrored variant: concatenation of A034869 and A014462.

			.   0:                                1
.   1:                              1   1
.   2:                            1   1   2
.   3:                          3   1   1   3
.   4:                        4   1   1   4   6
.   5:                     10   5   1   1   5   10
.   6:                   15   6   1   1   6   15  20
.   7:                 35   21  7   1   1   7   21   35
.   8:              56   28   8   1   1   8   28  56   70
.   9:           126   84   36  9   1   1   9   36   84   126
.  10:        210   120  45  10   1   1   10  45  120  210  252
.  11:     462   330  165   55  11  1   1   11  55  165   330  462
.  12:  792   495  220   66  12   1   1   12  66  220  495  792   924  .

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
Index entries for triangles and arrays related to Pascal's triangle.

Cf. A014413, A014462, A034868, A034869, A007318, A001405, A037952, A000079 (row sums), A001142 (row products).

a265848 n k = a265848_tabl !! n !! k
a265848_row n = a265848_tabl !! n
a265848_tabl = zipWith (++) ([] : a014413_tabf) a034868_tabf

row[n_] := Binomial[n, Join[Range[Floor[n/2] + 1, n], Range[0, Floor[n/2]]]]; Array[row, 12, 0] // Flatten (* Amiram Eldar, May 13 2025 *)

T(n,k) = A007318(n, (k + floor((n+2)/2)) mod (n+1)).
T(n,k) = if k <= [(n+1)/2] then A014413(n,k+1) else A034868(n,k-[(n+1)/2]).
T(n,0) = A037952(n) for n > 0.
T(n,n) = A001405(n).

A133179 A modular binomial sum transform of 2^n .

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 5, 15, 1, 3, 5, 15, 17, 51, 85, 255, 1, 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 1, 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535

Offset: 0

Philippe Deléham, Oct 10 2007

			A034868 is:
1;
1;
1, 2;
1, 3;
1, 4, 6;
1, 5, 10 ;...
A034868 modulo 2:
1;
1;
1, 0;
1, 1;
1, 0, 0;
1, 1, 0 ;...
a(0)=1*2^0 = 1;
a(1)=1*2^0 = 1;
a(2)=1*2^0+0*2^1 = 1;
a(3)=1*2^0+1*2^1 = 3;
a(4)=1*2^0+0*2^1+0*2^2 = 1
a(5)=1*2^0+1*2^1+0*2^2 = 3

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A034868, A048896, A101692, A130047.

A133179[n_] := Sum[2^k*Mod[Binomial[n, k], 2], {k, 0, Floor[n/2]}]; Table[A133179[n], {n,0,50}] (* G. C. Greubel, Aug 11 2017 *)

a(n) = Sum_{k=0..floor(n/2)} mod(binomial(n,k),2) * 2^k.

A258193 Define a<+>b = odd part(odd part(a) + odd part(b)), where odd part(n) = A000265(n); a(n) is the smallest prime of the form <+>_{0<=i<=k} binomial (n,i), or a(n)=0 if there is no such a prime (see comment).

Original entry on oeis.org

0, 0, 0, 0, 3, 3, 11, 0, 5, 3, 3, 17, 7, 23, 53, 29, 13, 5, 5, 3, 11, 3, 3, 823, 13, 7, 7, 457, 109, 109, 233, 2267, 17, 59, 151, 5, 19, 5, 5, 3, 113, 11, 11, 3, 23, 3, 3, 71, 43, 13, 13, 7, 179, 7, 7, 193, 29, 2137, 863, 443, 31, 5498157739, 977, 163

Offset: 1

Vladimir Shevelev, May 23 2015

nonn

f(n)=<+>_{0<=i<=n} c(i) is defined as the following: f(0)=c(0), f(n)=f(n-1)<+>c(n).

a(n)=0 for 1,2,3,4,8,82,107,...(cf. A258194)

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000265, A007318, A258126, A258127, A258194.

Cf. A034868, A010051.

import Data.Function (on)
a258193 n = head $ (filter ((== 1) . a010051'') $
                    scanl1 (<+>) (a034868_row n)) ++ [0]
            where (<+>) = (a000265 .) . on (+) a000265
-- Reinhard Zumkeller, Jun 20 2015

vSum[a_,b_]:=#[#[a]+#[b]]&[#/2^IntegerExponent[#,2]&];
Table[
First[Select[FoldList[vSum,First[#],Rest[#]]&[Map[Binomial[n,#]&,Range[0,n]]],PrimeQ]/.{}->{0}],{n,100}] (*Peter J. C. Moses, May 23 2015*)

More terms from Peter J. C. Moses, May 23 2015

A225860 Number of partitions of 2^n into binomial coefficients C(n,k).

Original entry on oeis.org

1, 1, 3, 3, 10, 16, 55, 133, 599, 1956, 11982, 57872, 477289, 3290993, 37671322, 373566217, 5986589127, 85738839408, 1931359427404, 40003346563574, 1274368885871702, 38222180804346119, 1729302096638372691, 75195441157176495562, 4848355840082055530710

Offset: 0

Reinhard Zumkeller, May 26 2013

nonn

			n=3: C(3,0)=C(3,3) = 1, C(3,1)=C(3,2) = 3, 2^3 = 8:
a(3) = #{3+3+1+1, 3+1+1+1+1+1, 8x1} = 3;
n=4: C(4,0)=C(4,4) = 1, C(4,1)=C(4,3) = 4, C(4,2) = 6, 2^4 = 16:
a(4) = #{6+6+4, 6+6+1+1+1+1, 6+4+4+1+1, 6+4+6x1, 6+10x1, 4+4+4+4, 4+4+4+1+1+1+1, 4+4+8x1, 4+12x1, 16x1} = 10
n=5: C(5,0)=C(5,5) = 1, C(5,1)=C(5,4) = 5, C(5,2)=C(5,3) = 10, 2^5 = 32:
a(5) = #{10+10+10+1+1, 10+10+5+5+1+1, 10+10+5+7x1, 10+10+12x1, 10+5+5+5+5+1+1, 10+5+5+5+7x1, 10+5+5+12x1, 10+5+17x1, 10+22x1, 6x5, 5x5+7x1, 5+5+5+5+12x1, 5+5+5+17x1, 5+5+22x1, 5+27x1, 32x1} = 16.

Cf. A000079, A034868, A007318.

a225860 n = p (a034868_row n) (2 ^ n) where
   p _          0 = 1
   p []         _ = 0
   p bs'@(b:bs) m = if m < b then 0 else p bs' (m - b) + p bs m

a:= proc(n) option remember; local g, i, j, l, m, t;
      m:= 1+iquo(n, 2);
      l:= array(1..m, [seq(binomial(n,k), k=0..m-1)]);
      g:= array(1..m, [seq(array(0..l[i]-1, [0$(l[i])]), i=1..m)]);
      g[1][0]:= 1;
      for j from 0 to 2^n do for i from 2 to m do
        g[i][irem(j, l[i])]:= g[i][irem(j, l[i])]
                             +g[i-1][irem(j, l[i-1])]
      od od; g[m][irem(2^n, l[m])]
    end:
seq(a(n), n=0..14);  # Alois P. Heinz, May 30 2013

a(13)-a(23) from Alois P. Heinz, May 30 2013

a(24) from Alois P. Heinz, Oct 06 2014

A273098 Discriminator of first half of row n of Pascal's triangle.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 4, 8, 13, 11, 13, 8, 17, 20, 12, 16, 23, 19, 35, 28, 41, 23, 41, 20, 37, 53, 47, 29, 61, 31, 76, 32, 63, 67, 61, 37, 101, 73, 131, 41, 101, 43, 67, 108, 83, 47, 119, 70, 137, 61, 118, 53, 127, 113, 137, 128, 167, 59, 179, 61, 173, 202, 206, 64, 239, 67, 334, 278, 213, 71, 179, 73, 185, 386, 269, 218, 277, 79, 197, 283

Offset: 0

Jeffrey Shallit, May 15 2016

nonn

The discriminator of a finite sequence is the least positive integer k such that all of its terms are pairwise incongruent, modulo k. Here the n-th term of the sequence is the discriminator of binomial(n, 0), binomial(n,1), ..., binomial(n,t) where t = floor(n/2).

It appears that a(2^k-1) = 2^k for k >= 3. - Robert Israel, May 15 2016

Robert Israel, Table of n, a(n) for n = 0..1000

Cf. A007318, A034868.

discriminator:= proc(L)
  local n,k;
  n:= nops(L);
  for k from n do if nops (L mod k) =n  then return k fi od:
end proc;
seq(discriminator({seq(binomial(n,j),j=0..floor(n/2))}),n=0..100); # Robert Israel, May 15 2016

cp's OEIS Frontend

A265848 Pascal's triangle, right and left halves interchanged.

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A133179 A modular binomial sum transform of 2^n .

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A258193 Define a<+>b = odd part(odd part(a) + odd part(b)), where odd part(n) = A000265(n); a(n) is the smallest prime of the form <+>_{0<=i<=k} binomial (n,i), or a(n)=0 if there is no such a prime (see comment).

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A225860 Number of partitions of 2^n into binomial coefficients C(n,k).

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A273098 Discriminator of first half of row n of Pascal's triangle.

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Maple