The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A166556 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A166556

Triangle read by rows, A000012 * A047999.
(history; published version)

#14 by OEIS Server at Tue Dec 03 08:24:34 EST 2024

LINKS

G. C. Greubel, <a href="/A166556/b166556_1.txt">Rows n = 0..100 of the triangle, flattened</a>

#13 by Michel Marcus at Tue Dec 03 08:24:34 EST 2024

STATUS

reviewed

approved

Discussion

Tue Dec 03

08:24

OEIS Server: Installed first b-file as b166556.txt.

#12 by Joerg Arndt at Tue Dec 03 04:29:22 EST 2024

STATUS

proposed

reviewed

#11 by G. C. Greubel at Tue Dec 03 01:41:01 EST 2024

STATUS

editing

proposed

#10 by G. C. Greubel at Tue Dec 03 01:40:46 EST 2024

NAME

Triangle read by rows, A000012 * A047999.

LINKS

G. C. Greubel, <a href="/A166556/b166556_1.txt">Rows n = 0..100 of the triangle, flattened</a>

FORMULA

Sum_{k=0..n} T(n, k0) = A006046A000027(n+1). (End)

T(n, 1) = A004526(n+1).

T(n, 2) = A004524(n+1).

T(2*n, n) = A080100(n).

Sum_{k=0..n} T(n, k) = A006046(n+1).

Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*( (1+(-1)^n))*A006046((n+4)/2) + (1-(-1)^n)*A006046((n+3)/2) ).

Sum_{k=0..floor(n/2)} T(n-k, k) = A007729(n). (End)

MATHEMATICA

A166556[n_, k_]:= Sum[Mod[Binomial[j, k], 2], {j, k, n}];

Table[A166556[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 02 2024 *)

PROG

(Magma)

A166556:= func< n, k | (&+[(Binomial(j, k) mod 2): j in [k..n]]) >;

[A166556(n, k): k in [0..n], n in [0..15]]; // G. C. Greubel, Dec 02 2024

(Python)

def A166556(n, k): return sum(binomial(j, k)%2 for j in range(k, n+1))

print(flatten([[A166556(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Dec 02 2024

CROSSREFS

Cf. A000027, A004524, A004526, A047999, A006046 (row sums)A080100.

Sums include: A006046 (row), A007729 (diagonal).

#9 by G. C. Greubel at Mon Dec 02 04:10:27 EST 2024

FORMULA

From G. C. Greubel, Dec 02 2024: (Start)

T(n, k) = Sum_{j=k..n} (binomial(j,k) mod 2).

Sum_{k=0..n} T(n, k) = A006046(n+1). (End)

EXAMPLE

. 1;

. 2, 1;

. 3, 1, 1;

. 4, 2, 2, 1;

. 5, 2, 2, 1, 1;

. 6, 3, 2, 1, 2, 1;

. 7, 3, 3, 1, 3, 1, 1;

. 8, 4, 4, 2, 4, 2, 2, 1;

. 9, 4, 4, 2, 4, 2, 2, 1, 1;

10, 5, 4, 2, 4, 2, 2, 1, 2, 1;

11, 5, 5, 2, 4, 2, 2, 1, 3, 1, 1;

12, 6, 6, 3, 4, 2, 2, 1, 4, 2, 2, 1;

13, 6, 6, 3, 5, 2, 2, 1, 5, 2, 2, 1, 1;

...

STATUS

approved

editing

#8 by R. J. Mathar at Thu Jul 21 11:39:38 EDT 2016

STATUS

editing

approved

#7 by R. J. Mathar at Thu Jul 21 11:39:21 EDT 2016

COMMENTS

Sum of n-th row terms = (1, 3, 5, 11, 15, 19, 27, 29,...) = A006046(n+1)

MAPLE

A166556 := proc(n, k)

local j;

add(A047999(j, k), j=k..n) ;

end proc: # R. J. Mathar, Jul 21 2016

CROSSREFS

Cf. A047999, A006046 (row sums).

KEYWORD

nonn,easy,tabl

STATUS

approved

editing

#6 by Joerg Arndt at Sat Jan 05 09:07:58 EST 2013

STATUS

proposed

approved

#5 by Michel Marcus at Sat Jan 05 08:59:07 EST 2013

STATUS

editing

proposed