A156348 -id:A156348 - cp's OEIS frontend

A156834 A156348 * A000010.

1, 2, 3, 5, 5, 12, 7, 17, 19, 30, 11, 63, 13, 56, 99, 89, 17, 154, 19, 269, 237, 132, 23, 509, 301, 182, 379, 783, 29, 1230, 31, 881, 813, 306, 2125, 2431, 37, 380, 1299, 4157, 41, 4822, 43, 3695, 6175, 552, 47, 8529, 5587, 6266, 2787

Offset: 1

Gary W. Adamson, Feb 16 2009

Conjecture: for n>1, a(n) = n iff n is prime. Companion to A156833.

			a(4) = 5 = (1, 2, 0, 1) dot (1, 1, 2, 2) = (1 + 2 + 0 + 2), where row 4 of A156348 = (1, 2, 0, 1) and (1, 1, 2, 2) = the first 4 terms of Euler's phi function.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A156348, A000010, A156833, A157020, A343553.

Equals row sums of triangle A157030. [Gary W. Adamson, Feb 21 2009]

A156834 := proc(n)
        add(A156348(n,k)*numtheory[phi](k),k=1..n) ;
end proc: # R. J. Mathar, Mar 03 2013

a[n_] := DivisorSum[n, EulerPhi[#] * Binomial[# + n/# - 2, #-1] &]; Array[a, 100] (* Amiram Eldar, Apr 22 2021 *)

a(n) = sumdiv(n, d, eulerphi(d)*binomial(d+n/d-2, d-1)); \\ Seiichi Manyama, Apr 22 2021

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*(x/(1-x^k))^k)) \\ Seiichi Manyama, Apr 22 2021

Equals A156348 * A054525 * [1, 2, 3,...]; where A054525 = the inverse Mobius transform.
a(n) = Sum_{d|n} phi(d) * binomial(d+n/d-2, d-1). - Seiichi Manyama, Apr 22 2021
G.f.: Sum_{k >= 1} phi(k) * (x/(1 - x^k))^k. - Seiichi Manyama, Apr 22 2021

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

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

A156837 Triangle read by rows, A051731 * A156348.

Original entry on oeis.org

1, 2, 1, 2, 0, 1, 3, 3, 0, 1, 2, 0, 0, 0, 1, 4, 4, 4, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 4, 7, 0, 5, 0, 0, 0, 1, 3, 0, 7, 0, 0, 0, 0, 0, 1, 4, 6, 0, 0, 6, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 12, 14, 11, 0, 7, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson & Mats Granvik, Feb 16 2009

Left border: A000005(n).

Row sums = 3 iff n-th row is prime.

			First few rows of the triangle =
1;
2, 1;
2, 0, 1;
3, 3, 0, 1;
2, 0, 0, 0, 1;
4, 4, 4, 0, 0, 1;
2, 0, 0, 0, 0, 0, 1;
4, 7, 0, 5, 0, 0, 0, 1;
3, 0, 7, 0, 0, 0, 0, 0, 1;
4, 6, 0, 0, 6, 0, 0, 0, 0, 1;
2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
6, 12, 14, 11, 0, 7, 0, 0, 0, 0, 0, 1;
2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
4, 8, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 1;
...

Cf. A156348, A051731, A000005, A156838 (row sums).

A156837 := proc(n,k)
    add(A051731(n,j)*A156348(j,k),j=k..n) ;
end proc: # R. J. Mathar, Mar 03 2013

Triangle read by rows, A051731 * A156348 = inverse Mobius transform of A156348

A157497 Triangle read by rows, A156348 * A127648.

Original entry on oeis.org

1, 1, 2, 1, 0, 3, 1, 4, 0, 4, 1, 0, 0, 0, 5, 1, 6, 9, 0, 0, 6, 1, 0, 0, 0, 0, 0, 7, 1, 8, 0, 16, 0, 0, 0, 8, 1, 0, 18, 0, 0, 0, 0, 0, 9, 1, 10, 0, 0, 25, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 1, 12, 30, 40, 0, 36, 0, 0, 0, 0, 0, 12

Offset: 1

Gary W. Adamson & Mats Granvik, Mar 01 2009

Row sums = A157020: (1, 3, 4, 9, 6, 22, 8,...)

			First few rows of the triangle =
1;
1, 2;
1, 0, 3;
1, 4, 0, 4;
1, 0, 0, 0, 5;
1, 6, 9, 0, 0, 6;
1, 0, 0, 0, 0, 0, 7;
1, 8, 0, 16, 0, 0, 0, 8;
1, 0, 18, 0, 0, 0, 0, 0, 9;
1, 10, 0, 0, 25, 0, 0, 0, 0, 10;
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11;
1, 12, 30, 40, 0, 36, 0, 0, 0, 0, 0, 12;
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13;
1, 14, 0, 0, 0, 0, 49, 0, 0, 0, 0, 0, 0, 14;
...
Row 4 = (1, 4, 0, 4) = termwise products of (1, 2, 0, 1) and (1, 2, 3, 4)
where (1, 2, 0, 1) = row 4 of triangle A156348.

Cf. A156348, A126648, A156020

Triangle read by rows, A156348 * A127648. A127648 = an infinite lower triangular matrix with (1, 2, 3,...) as the main diagonal and the rest zeros.

A157028 Triangle read by rows, A007318 * A156348.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 8, 5, 3, 1, 16, 12, 6, 4, 1, 32, 28, 13, 10, 5, 1, 64, 64, 33, 20, 15, 6, 1, 128, 144, 84, 39, 35, 21, 7, 1, 256, 320, 202, 88, 70, 56, 28, 8, 1, 512, 704, 468, 228, 131, 126, 84, 36, 9, 1, 1024, 1536, 1071, 600, 260, 252, 210, 120, 45, 10, 1

Offset: 1

Gary W. Adamson & Mats Granvik, Feb 21 2009

Row sums = A157029: (1, 3, 7, 17, 39, 89, 203, 459,...).

			1;
2, 1;
4, 2, 1;
8, 5, 3, 1;
16, 12, 6, 4, 1;
32, 28, 13, 10, 5, 1;
64, 64, 33, 20, 15, 6, 1;
128, 144, 84, 39, 35, 21, 7, 1;
256, 320, 202, 88, 70, 56, 28, 8, 1;
512, 704, 468, 228, 131, 126, 84, 36, 9, 1;
1024, 1536, 1071, 600, 260, 252, 210, 120, 45, 10, 1;
2048, 3328, 2441, 1495, 605, 468, 462, 330, 165, 55, 11, 1;
4096, 7168, 5532, 3508, 1595, 864, 924, 792, 495, 220, 66, 12, 1;
...

Cf. A156348, A157019, A157029

Triangle read by rows, binomial transform of A156348.

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.

A156839 Triangle read by rows, A054525 * A156348 * A000012.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 5, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 6, 4, 4, 1, 1, 1, 1, 6, 6, 6, 1, 1, 1, 1, 1, 1, 1, 9, 9, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 24, 24, 22, 15, 6, 6, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson and Mats Granvik, Feb 16 2009

For rows >1, n-th row = all 1's iff n is prime.

Row sums = A156833: (1, 2, 3, 6, 5, 16, 7, 24, 24, 38, 11,...).

			First few rows of the triangle =
1;
1, 1;
1, 1, 1;
2, 2, 1, 1;
1, 1, 1, 1, 1;
5, 5, 3, 1, 1, 1;
1, 1, 1, 1, 1, 1, 1;
6, 6, 4, 4, 1, 1, 1, 1;
6, 6, 6, 1, 1, 1, 1, 1, 1;
9, 9, 5, 5, 5, 1, 1, 1, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
24, 24, 22, 15, 6, 6, 1, 1, 1, 1, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
13, 13, 7, 7, 7, 7, 7, 1, 1, 1, 1, 1, 1, 1;
...

Cf. A156348, A054525, A156833

Triangle read by rows, A054525 * A156348 * A000012 A054525 = the inverse Mobius transform, A000012 = an infinite lower triangular matrix with all 1's.

A157019 a(n) = Sum_{d|n} binomial(n/d+d-2, d-1).

Original entry on oeis.org

1, 2, 2, 4, 2, 8, 2, 10, 8, 12, 2, 34, 2, 16, 32, 38, 2, 62, 2, 92, 58, 24, 2, 210, 72, 28, 92, 198, 2, 394, 2, 274, 134, 36, 422, 776, 2, 40, 184, 1142, 2, 1178, 2, 618, 1232, 48, 2, 2634, 926, 1482, 308, 964, 2, 2972, 2004, 4610, 382, 60, 2, 8576, 2, 64, 6470, 5130

Offset: 1

R. J. Mathar, Feb 21 2009

Equals row sums of triangle A156348. - Gary W. Adamson & Mats Granvik, Feb 21 2009
a(n) = 2 iff n is prime.
The binomial transform (note the offset) is 0, 1, 4, 11, 28, 67, 156, 359, 818, 1847, 4146, 9275, ... - R. J. Mathar, Mar 03 2013
a(n) is the number of distinct paths that connect the starting (1,1) point to the hyperbola with equation (x * y = n), when the choice for a move is constrained to belong to { (x := x + 1), (y := y + 1) }. - Luc Rousseau, Jun 27 2017

			a(4) = 4 = 1 + 2 + 0 + 1.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Paul D. Hanna)

Cf. A081543, A018818, A156838 (Mobius transform).
Cf. A156348.
Cf. A000010.

A157019 := proc(n) add( binomial(n/d+d-2, d-1), d=numtheory[divisors](n) ) ; end:

a[n_] := Sum[Binomial[n/d + d - 2, d - 1], {d, Divisors[n]}];
Array[a, 100] (* Jean-François Alcover, Mar 24 2020 *)

{a(n)=polcoeff(sum(m=1,n,x^m/(1-x^m+x*O(x^n))^m),n)} \\ Paul D. Hanna, Mar 01 2009

G.f.: A(x) = Sum_{n>=1} x^n/(1 - x^n)^n. - Paul D. Hanna, Mar 01 2009

a(n) = Sum_{k=1..n} binomial(gcd(n,k) + n/gcd(n,k) - 2, gcd(n,k) - 1) / phi(n/gcd(n,k)) = Sum_{k=1..n} binomial(gcd(n,k) + n/gcd(n,k) - 2, n/gcd(n,k) - 1) / phi(n/gcd(n,k)) where phi = A000010. - Richard L. Ollerton, May 19 2021

A156838 Row sums of triangle A156837.

Original entry on oeis.org

1, 3, 3, 7, 3, 13, 3, 17, 11, 17, 3, 51, 3, 21, 37, 55, 3, 83, 3, 113, 63, 29, 3, 271, 75, 33, 103, 223, 3, 453, 3, 329, 139, 41, 427, 897, 3, 45, 189, 1265, 3, 1267, 3, 651, 1277, 53, 3, 2943, 929, 1571, 313, 1001, 3, 3147, 2009, 4843, 387, 65, 3, 9159, 3, 69, 6541, 5459, 3647, 6753, 3, 2053, 559, 15299, 3, 25677, 3, 81

Offset: 1

Gary W. Adamson & Mats Granvik, Feb 16 2009

a(n) = 3 iff n is prime.

			a(4) = 7 since row 4 of triangle A156837 = (3, 3, 0, 1).

Antti Karttunen, Table of n, a(n) for n = 1..10000

A156837

A156838 := proc(n)
    add(A156837(n,k),k=1..n) ;
end proc: # R. J. Mathar, Mar 03 2013

A156348(n, k) = if(k<1 || k>n || n%k, 0, binomial(n/k-2+k, k-1)); \\ After R. J. Mathar's Maple-program in A156348
A156837(n, k) = sum(j=k,n,if(n%j, 0, A156348(j, k)));
A156838(n) = sumdiv(n,d,A156837(n,d)); \\ Antti Karttunen, Nov 30 2024

Row sums of triangle A156837.

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

More terms from Antti Karttunen, Nov 30 2024

A157029 A007318 * A157019.

Original entry on oeis.org

1, 3, 7, 17, 39, 89, 203, 459, 1029, 2299, 5129, 11409, 25273, 55787, 122875, 270239, 593331, 1299883, 2841243, 6197855, 13499235, 29366411, 63809311, 138466835, 300036895, 649186659, 1402796793, 3027908077, 6529611587, 14068804905

Offset: 1

Gary W. Adamson & Mats Granvik, Feb 21 2009

nonn

Equals row sums of triangle A157028.

			a(4) = 17 = (1, 3, 3, 1) dot (1, 2, 2, 4) = (1 + 6 + 6 + 4). a(4) = 17 = sum of row 4 terms, triangle A157028: (8 + 5 + 3 + 1).
G.f.: A(x) = x + 3*x^2 + 7*x^3 + 17*x^4 + 39*x^5 + 89*x^6 + 203*x^7 + 459*x^8 + 1029*x^9 + 2299*x^10 + ...
such that
A(x) = x/((1-x) - x) + x^2*(1-x)^2/((1-x)^2 - x^2)^2 + x^3*(1-x)^6/((1-x)^3 - x^3)^3 + x^4*(1-x)^12/((1-x)^4 - x^4)^4 + x^5*(1-x)^20/((1-x)^5 - x^5)^5 + ...

Cf. A157019, A157028, A156348

G.f.: Sum_{n>=1} x^n * (1-x)^(n*(n-1)) / ((1-x)^n - x^n)^n. - Paul D. Hanna, Mar 26 2018

G.f.: Sum_{n>=1} x^n/(1-x)^n / (1 - x^n/(1-x)^n)^n. - Paul D. Hanna, Mar 26 2018

Extended by R. J. Mathar, Apr 07 2009

cp's OEIS Frontend

A156834 A156348 * A000010.

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

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A156837 Triangle read by rows, A051731 * A156348.

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A157497 Triangle read by rows, A156348 * A127648.

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A157028 Triangle read by rows, A007318 * A156348.

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A156836 Triangle read by rows, A156348 * A130207.

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A156839 Triangle read by rows, A054525 * A156348 * A000012.

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A157019 a(n) = Sum_{d|n} binomial(n/d+d-2, d-1).

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