A157019 -id:A157019 - cp's OEIS frontend

A157029 A007318 * A157019.

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

A157031 Triangle A054521 * A157019, where A054521 = an infinite lower triangular matrix and A157019 = a vector [1, 2, 2, 4, 2, 8, 2, 10, 8, ...].

Original entry on oeis.org

1, 1, 3, 3, 9, 3, 19, 7, 21, 13, 51, 7, 87, 17, 39, 51, 175, 11, 239, 21, 169, 111, 415, 15, 489, 185, 313, 219, 1017, 15, 1413, 283, 763, 415, 981, 513, 3057, 839, 1259, 497, 4425, 93, 5605, 893, 1311, 2259, 7505, 521, 8267, 1429, 5473, 3311, 13821, 1449, 11135, 4095

Offset: 1

Gary W. Adamson & Mats Granvik, Feb 21 2009

nonn

			a(4) = 3 = (1, 0, 1, 0) dot (1, 2, 2, 4) = (1 + 0 + 2 + 0), where (1, 0, 1, 0) equals row 4 of triangle A054521.

Cf. A157019, A054521.

A157031 := proc(n)
    add(A054521(n,k)*A157019(k),k=1..n) ;
end proc: # R. J. Mathar, Jul 21 2015

A081543 G.f.: Sum_{k >= 1} x^k/(1-x^k)^(k+1).

Original entry on oeis.org

1, 3, 4, 8, 6, 17, 8, 24, 20, 32, 12, 76, 14, 51, 72, 97, 18, 158, 20, 213, 142, 101, 24, 491, 152, 132, 248, 479, 30, 915, 32, 681, 398, 206, 828, 1859, 38, 249, 600, 2560, 42, 2692, 44, 1686, 2864, 347, 48, 6166, 1766, 3405, 1192, 2811, 54, 6796, 4424, 9987

Offset: 1

Benoit Cloitre, Apr 21 2003

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..999 from Harvey P. Dale)

Cf. A157019.

With[{nn=50},CoefficientList[Series[Sum[x^k/(1-x^k)^(k+1),{k,nn}],{x,0,nn}],x]] (* Harvey P. Dale, May 28 2017 *)

a(n)=if(n<1,0,polcoeff(sum(k=1,n,1/(1-x^k)^k,x*O(x^(n^2))),n))

If p is prime then a(p)=p+1.

a(n) = Sum_{d|n} binomial(d-1+n/d,d). - R. J. Mathar, Feb 21 2009

Description corrected by Vladeta Jovovic, Aug 22 2003

Corrected offset R. J. Mathar, Feb 21 2009

A324158 Expansion of Sum_{k>=1} x^k / (1 - k * x^k)^k.

Original entry on oeis.org

1, 2, 2, 6, 2, 23, 2, 50, 56, 107, 2, 660, 2, 499, 1592, 2370, 2, 8246, 2, 18557, 21786, 11387, 2, 175198, 43752, 53419, 298892, 487762, 2, 1891098, 2, 2552066, 3905222, 1114403, 3785462, 29081597, 2, 4981099, 48376512, 95510772, 2, 218764940, 2, 346411232, 770590352

Offset: 1

Ilya Gutkovskiy, Sep 02 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A055225, A087909, A157019, A157020, A167531, A324159.

nmax = 45; CoefficientList[Series[Sum[x^k/(1 - k x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[(n/d)^(d - 1) Binomial[n/d + d - 2, d - 1], {d, Divisors[n]}], {n, 1, 45}]

a(n) = sumdiv(n, d, (n/d)^(d-1) * binomial(n/d+d-2, d-1)); \\ Michel Marcus, Sep 02 2019

N=66; x='x+O('x^N); Vec(sum(k=1, N, x^k/(1-k*x^k)^k)) \\ Seiichi Manyama, Sep 03 2019

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

a(p) = 2, where p is prime.

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 *)

A324159 Expansion of Sum_{k>=1} k * x^k / (1 - k * x^k)^k.

Original entry on oeis.org

1, 3, 4, 13, 6, 58, 8, 137, 172, 296, 12, 2063, 14, 1254, 5536, 7697, 18, 25201, 20, 68976, 70862, 23882, 24, 607485, 218776, 108720, 918568, 1810089, 30, 6746147, 32, 9408545, 11779582, 2233172, 19935756, 102405280, 38, 9968370, 145283360, 393585971, 42, 730233631, 44, 1296043651, 2718300016

Offset: 1

Ilya Gutkovskiy, Sep 02 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A055225, A087909, A157019, A157020, A324158.

nmax = 45; CoefficientList[Series[Sum[k x^k/(1 - k x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[(n/d)^d Binomial[n/d + d - 2, d - 1], {d, Divisors[n]}], {n, 1, 45}]

a(n) = sumdiv(n, d, (n/d)^d * binomial(n/d+d-2, d-1)); \\ Michel Marcus, Sep 02 2019

N=66; x='x+O('x^N); Vec(sum(k=1, N, k*x^k/(1-k*x^k)^k)) \\ Seiichi Manyama, Sep 03 2019

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

a(p) = p + 1, where p is prime.

A338661 a(n) = Sum_{d|n} d^n * binomial(d+n/d-2, d-1).

Original entry on oeis.org

1, 5, 28, 289, 3126, 49036, 823544, 17040385, 387538588, 10048833246, 285311670612, 8929334253419, 302875106592254, 11116754387182648, 437894348359764856, 18448995959423107073, 827240261886336764178, 39347761059781438793815, 1978419655660313589123980

Offset: 1

Seiichi Manyama, Apr 22 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..386

Cf. A023887, A157019, A157020, A324158, A324159, A339481, A339482, A339712, A343573.

a[n_] := DivisorSum[n, #^n * Binomial[# + n/# - 2, #-1] &]; Array[a, 20] (* Amiram Eldar, Apr 22 2021 *)

a(n) = sumdiv(n, d, d^n*binomial(d+n/d-2, d-1));

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x/(1-(k*x)^k))^k))

G.f.: Sum_{k >= 1} (k * x/(1 - (k * x)^k))^k.
If p is prime, a(p) = 1 + p^p.
a(n) ~ n^n. - Vaclav Kotesovec, Aug 04 2025

A343573 a(n) = Sum_{d|n} d^d * binomial(d+n/d-2, d-1).

Original entry on oeis.org

1, 5, 28, 265, 3126, 46750, 823544, 16778257, 387420652, 10000015646, 285311670612, 8916100731047, 302875106592254, 11112006831322846, 437893890380906656, 18446744073843774497, 827240261886336764178, 39346408075300025340205

Offset: 1

Seiichi Manyama, Apr 20 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..386

Cf. A023887, A157019, A157020, A324158, A324159, A338661, A339481, A339482, A339712, A343567, A343574.

a[n_] := DivisorSum[n, #^#*Binomial[# + n/# - 2, # - 1] &]; Array[a, 20] (* Amiram Eldar, Apr 20 2021 *)

a(n) = sumdiv(n, d, d^d*binomial(d+n/d-2, d-1));

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x/(1-x^k))^k))

G.f.: Sum_{k >= 1} (k * x/(1 - x^k))^k.

If p is prime, a(p) = 1 + p^p.

A339481 a(n) = Sum_{d|n} d^(n-d) * binomial(d+n/d-2, d-1).

Original entry on oeis.org

1, 2, 2, 10, 2, 131, 2, 1282, 4376, 16907, 2, 1138272, 2, 5793475, 154455992, 469893122, 2, 49501130330, 2, 1318441711177, 19001093813466, 3138439911059, 2, 15989399214596398, 6675720214843752, 3937376603803099, 6754271297694102092, 47097064577536888014, 2

Offset: 1

Seiichi Manyama, Apr 24 2021

nonn

Cf. A157019, A157020, A324158, A324159, A338661, A339482, A339712, A343573.

a[n_] := DivisorSum[n, #^(n - #) * Binomial[# + n/# - 2, # - 1] &]; Array[a, 30] (* Amiram Eldar, Apr 25 2021 *)

a(n) = sumdiv(n, d, d^(n-d)*binomial(d+n/d-2, d-1));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (x/(1-(k*x)^k))^k))

G.f.: Sum_{k >= 1} (x/(1 - (k * x)^k))^k.

If p is prime, a(p) = 2.

A217670 G.f.: Sum_{n>=0} x^n/(1 + x^n)^n.

Original entry on oeis.org

1, 1, 0, 2, -2, 2, 0, 2, -8, 8, 0, 2, -12, 2, 0, 32, -36, 2, 0, 2, -20, 58, 0, 2, -136, 72, 0, 92, -28, 2, 0, 2, -272, 134, 0, 422, -288, 2, 0, 184, -480, 2, 0, 2, -44, 1232, 0, 2, -2360, 926, 0, 308, -52, 2, 0, 2004, -1176, 382, 0, 2, -4064, 2, 0, 6470, -5128, 3642

Offset: 0

Paul D. Hanna, Oct 10 2012

sign

			G.f.: A(x) = 1 + x + 2*x^3 - 2*x^4 + 2*x^5 + 2*x^7 - 8*x^8 + 8*x^9 +...
where
A(x) = 1 + x/(1+x) + x^2/(1+x^2)^2 + x^3/(1+x^3)^3 + x^4/(1+x^4)^4 + x^5/(1+x^5)^5 +...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Paul D. Hanna)

Cf. A048272, A157019, A217668.

terms = 100; Sum[x^n/(1 + x^n)^n, {n, 0, terms}] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, May 16 2017 *)

{a(n)=polcoeff(sum(m=0, n, x^m/(1+x^m +x*O(x^n))^m), n)}
for(n=0, 100, print1(a(n), ", "))

a(n) = if(n==0, 1, sumdiv(n, d, (-1)^(d-1)*binomial(d+n/d-2, d-1))); \\ Seiichi Manyama, Apr 23 2021

a(4*n+2) = 0 for n>=0.
From Seiichi Manyama, Apr 23 2021: (Start)
a(n) = Sum_{d|n} (-1)^(d-1) * binomial(d+n/d-2, d-1) for n > 0.
If p is prime, a(p) = 1 + (-1)^(p-1). (End)

cp's OEIS Frontend

A157029 A007318 * A157019.

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A157031 Triangle A054521 * A157019, where A054521 = an infinite lower triangular matrix and A157019 = a vector [1, 2, 2, 4, 2, 8, 2, 10, 8, ...].

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A081543 G.f.: Sum_{k >= 1} x^k/(1-x^k)^(k+1).

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A324158 Expansion of Sum_{k>=1} x^k / (1 - k * x^k)^k.

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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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A324159 Expansion of Sum_{k>=1} k * x^k / (1 - k * x^k)^k.

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A338661 a(n) = Sum_{d|n} d^n * binomial(d+n/d-2, d-1).

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A343573 a(n) = Sum_{d|n} d^d * binomial(d+n/d-2, d-1).

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