A156834 -id:A156834 - cp's OEIS frontend

A157030 Triangle read by rows, A156834 * A054521.

1, 2, 0, 2, 1, 0, 4, 0, 1, 0, 2, 1, 1, 1, 0, 8, 3, 0, 0, 1, 0, 2, 1, 1, 1, 1, 1, 0, 10, 0, 5, 0, 1, 0, 1, 0, 8, 7, 0, 1, 1, 0, 1, 1, 0, 12, 5, 6, 5, 0, 0, 10, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 34, 10, 10, 0, 7, 0, 10, 0, 0, 1, 0

Offset: 1

Gary W. Adamson and Mats Granvik, Feb 21 2009

Left border = A157019: (1, 2, 2, 4, 2, 8, 2, 10,...).

			First few rows of the triangle =
1;
2, 0;
2, 1, 0;
4, 0, 1, 0;
2, 1, 1, 1, 0;
8, 3, 0, 0, 1, 0;
2, 1, 1, 1, 1, 1, 0;
10, 0, 5, 0, 1, 0, 1, 0;
8, 7, 0, 1, 1, 0, 1, 1, 0;
12, 5, 6, 5, 0, 0, 10, 1, 0;
2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0;
34, 10, 10, 0, 7, 0, 1, 0, 0, 0, 1, 0;
2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0;
...

Cf. A156834 (row sums), A054521.

Triangle read by rows, A156834 * A054521; as infinite lower triangular matrices.

A156348 Triangle T(n,k) read by rows. Column of Pascal's triangle interleaved with k-1 zeros.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 0, 0, 0, 1, 1, 3, 3, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 4, 0, 4, 0, 0, 0, 1, 1, 0, 6, 0, 0, 0, 0, 0, 1, 1, 5, 0, 0, 5, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 6, 10, 10, 0, 6, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 7, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0

Offset: 1

Mats Granvik, Feb 08 2009

The rows of the Pascal triangle are here found as square root parabolas like in the plots at www.divisorplot.com. Central binomial coefficients are found at the square root boundary.
A156348 * A000010 = A156834: (1, 2, 3, 5, 5, 12, 7, 17, 19, 30, 11, ...). - Gary W. Adamson, Feb 16 2009
Row sums give A157019.

			Table begins:
1
1  1
1  0  1
1  2  0  1
1  0  0  0  1
1  3  3  0  0  1
1  0  0  0  0  0  1
1  4  0  4  0  0  0  1
1  0  6  0  0  0  0  0  1
1  5  0  0  5  0  0  0  0  1
1  0  0  0  0  0  0  0  0  0  1
1  6 10 10  0  6  0  0  0  0  0  1
1  0  0  0  0  0  0  0  0  0  0  0  1
1  7  0  0  0  0  7  0  0  0  0  0  0  1
1  0 15  0 15  0  0  0  0  0  0  0  0  0  1
1  8  0 20  0  0  0  8  0  0  0  0  0  0  0  1
1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1
1  9 21  0  0 21  0  0  9  0  0  0  0  0  0  0  0  1
1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1
1 10  0 35 35  0  0  0  0 10  0  0  0  0  0  0  0  0  0  1

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
el Houcein el Abdalaoui, Mohamed Dahmoune and Djelloul Ziadi, On the transition reduction problem for finite automata, arXiv preprint arXiv:1301.3751 [cs.FL], 2013. - From _N. J. A. Sloane_, Feb 12 2013
Jeff Ventrella, Divisor Plot
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A051731,A156834.

Following Mathar's Maple program.
a156348 n k = a156348_tabl !! (n-1) !! (k-1)
a156348_tabl = map a156348_row [1..]
a156348_row n = map (f n) [1..n] where
   f n k = if r == 0 then a007318 (n' - 2 + k) (k - 1) else 0
           where (n', r) = divMod n k
-- Reinhard Zumkeller, Jan 31 2014

A156348 := proc(n,k)
    if k < 1 or k > n then
        return 0 ;
    elif n mod k = 0 then
        binomial(n/k-2+k,k-1) ;
    else
        0 ;
    end if;
end proc: # R. J. Mathar, Mar 03 2013

T[n_, k_] := Which[k < 1 || k > n, 0, Mod[n, k] == 0, Binomial[n/k - 2 + k, k - 1], True, 0];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 16 2017 *)

A343517 a(n) = Sum_{1 <= x_1 <= x_2 <= ... <= x_n <= n} gcd(x_1, x_2, ... , x_n, n).

Original entry on oeis.org

1, 4, 12, 42, 130, 506, 1722, 6622, 24426, 93427, 352726, 1359388, 5200312, 20097156, 77567064, 300787366, 1166803126, 4539197723, 17672631918, 68933307843, 269129530770, 1052113994340, 4116715363822, 16124224571368, 63205303313900, 247961973949536

Offset: 1

Seiichi Manyama, Apr 17 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..1650

Main diagonal of A343516.

Cf. A000010, A156834, A332508, A332517.

a[n_] := DivisorSum[n, EulerPhi[n/#] * Binomial[n + # - 1, n] &]; Array[a, 25] (* Amiram Eldar, Apr 18 2021 *)

a(n) = sumdiv(n, d, eulerphi(n/d)*binomial(d+n-1, n));

a(n) = Sum_{d|n} phi(n/d) * binomial(d+n-1, n).
a(n) = [x^n] Sum_{k >= 1} phi(k) * x^k/(1 - x^k)^(n+1).
a(n) ~ 2^(2*n - 1) / sqrt(Pi*n). - Vaclav Kotesovec, May 23 2021

A156833 A054525 * A156348 * [1,2,3,...].

Original entry on oeis.org

1, 2, 3, 6, 5, 16, 7, 24, 24, 38, 11, 103, 13, 68, 127, 144, 17, 261, 19, 404, 291, 152, 23, 994, 370, 206, 540, 1093, 29, 2195, 31, 1584, 943, 338, 2543, 4808, 37, 416, 1479, 7371, 41, 7929, 43, 4691, 8976, 596, 47, 18876, 6510, 11035, 3091

Offset: 1

Gary W. Adamson, Feb 16 2009

nonn

Conjecture: for n>1, a(n) = n iff n is prime.

Companion to A156834: (1, 2, 3, 5, 5, 12, 7, 17, 19,...).

			a(4) = 6 since first 4 terms of A156348 * [1, 2, 3, 4,...] = (1, 3, 4, 9);
Then (1, 3, 4, 9) dot (0, -1, 0, 1) = (0 - 3 + 0 + 9) = 6. Row 4 of A054525 = (0, -1, 0, 1).

Cf. A156348, A054525, A156834

A156833T := proc(n,k)
    add(A054525(n,j)*A156348(j,k),j=k..n) ;
end proc:
A156833 := proc(n)
    add(A156833T(n,k)*k,k=1..n) ;
end proc: # R. J. Mathar, Mar 03 2013

A054525[n_, k_] := If[Divisible[n, k], MoebiusMu[n/k], 0];
A156348[n_, k_] := Which[k < 1 || k > n, 0, Mod[n, k] == 0, Binomial[n/k - 2 + k, k - 1], True, 0];
T[n_, k_] := Sum[A054525[n, j]*A156348[j, k], {j, k, n}];
a[n_] := Sum[T[n, k]*k, {k, 1, n}];
Table[a[n], {n, 1, 51}] (* Jean-François Alcover, Oct 15 2023 *)

A054525 * A156348 * [1,2,3,...]

Extended beyond a(14) by R. J. Mathar, Mar 03 2013

A156836 Triangle read by rows, A156348 * A130207.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 2, 0, 2, 1, 0, 0, 0, 4, 1, 3, 6, 0, 0, 2, 1, 0, 0, 0, 0, 0, 6, 1, 4, 0, 8, 0, 0, 0, 4, 1, 0, 12, 0, 0, 0, 0, 0, 6, 1, 5, 0, 0, 20, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 1, 6, 20, 20, 0, 12, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12

Offset: 1

Gary W. Adamson & Mats Granvik, Feb 16 2009

Row sums = A156834: (1, 2, 3, 5, 5, 12, 7, 17, 19, 30, 11,...).

			First few rows of the triangle =
1;
1, 1;
1, 0, 2;
1, 2, 0, 2;
1, 0, 0, 0, 4;
1, 3, 6, 0, 0, 2;
1, 0, 0, 0, 0, 0, 6;
1, 4, 0, 8, 0, 0, 0, 4;
1, 0, 12, 0, 0, 0, 0, 0, 6;
1, 5, 0, 0, 20, 0, 0, 0, 0, 4;
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10;
1, 6, 20, 20, 0, 12, 0, 0, 0, 0, 0, 4;
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12;
1, 7, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 0, 6;
...

Cf. A156348, A130207, A156834, A000010

Triangle read by rows, A156348 * A130207, where A130207 = an infinite lower

triangular matrix with A000010 as the main diagonal and the rest zeros.

A338655 a(n) = Sum_{d|n} phi(d) * binomial(d+n/d-1, d).

Original entry on oeis.org

1, 3, 5, 9, 9, 22, 13, 32, 35, 53, 21, 121, 25, 96, 177, 166, 33, 297, 37, 491, 417, 218, 45, 1002, 549, 297, 705, 1375, 57, 2418, 61, 1640, 1405, 491, 3887, 4659, 73, 606, 2233, 8156, 81, 8989, 85, 6189, 11955, 872, 93, 16550, 10387, 12927, 4757, 11111, 105, 22392, 25757

Offset: 1

Seiichi Manyama, Apr 22 2021

nonn

Cf. A000010, A156834, A343517, A338658.

a[n_] := DivisorSum[n, EulerPhi[#] * Binomial[# + n/# - 1, #] &]; Array[a, 100] (* Amiram Eldar, Apr 22 2021 *)

a(n) = sumdiv(n, d, eulerphi(d)*binomial(d+n/d-1, d));

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k/(1-x^k)^(k+1)))

G.f.: Sum_{k >= 1} phi(k) * x^k/(1 - x^k)^(k+1).
If p is prime, a(p) = 2*p - 1.
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} binomial(gcd(n,k) + n/gcd(n,k) - 1,n/gcd(n,k)).
a(n) = Sum_{k=1..n} binomial(gcd(n,k) + n/gcd(n,k) - 1,gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

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A157030 Triangle read by rows, A156834 * A054521.

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A156348 Triangle T(n,k) read by rows. Column of Pascal's triangle interleaved with k-1 zeros.

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A343517 a(n) = Sum_{1 <= x_1 <= x_2 <= ... <= x_n <= n} gcd(x_1, x_2, ... , x_n, n).

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A156833 A054525 * A156348 * [1,2,3,...].

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A156836 Triangle read by rows, A156348 * A130207.

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A338655 a(n) = Sum_{d|n} phi(d) * binomial(d+n/d-1, d).

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