Search: a318637 -id:a318637
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A318636
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Expansion of Sum_{n>=1} ( (1 + x^n)^n - 1 ).
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+10
15
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1, 2, 3, 5, 5, 9, 7, 14, 10, 20, 11, 31, 13, 35, 25, 45, 17, 74, 19, 70, 56, 77, 23, 161, 26, 104, 111, 154, 29, 261, 31, 222, 198, 170, 56, 536, 37, 209, 325, 496, 41, 623, 43, 605, 626, 299, 47, 1407, 50, 602, 731, 1092, 53, 1305, 517, 1443, 1026, 464, 59, 4002, 61, 527, 1429, 2381, 1352, 2596, 67, 3009, 1840, 2787, 71, 6719, 73, 740, 5378, 4655, 407, 5135, 79, 10118
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OFFSET
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1,2
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LINKS
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FORMULA
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EXAMPLE
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G.f.: A(x) = x + 2*x^2 + 3*x^3 + 5*x^4 + 5*x^5 + 9*x^6 + 7*x^7 + 14*x^8 + 10*x^9 + 20*x^10 + 11*x^11 + 31*x^12 + 13*x^13 + 35*x^14 + 25*x^15 + 45*x^16 + ...
such that
A(x) = x + (1 + x^2)^2 - 1 + (1 + x^3)^3 - 1 + (1 + x^4)^4 - 1 + (1 + x^5)^5 - 1 + (1 + x^6)^6 - 1 + (1 + x^7)^7-1 + ...
RELATED SERIES.
The g.f. A(x) equals following series at y = 1:
Sum_{n>=1} ((y + x^n)^n - y^n) = x + 2*y*x^2 + 3*y^2*x^3 + (4*y^3 + 1)*x^4 + 5*y^4*x^5 + (6*y^5 + 3*y)*x^6 + 7*y^6*x^7 + (8*y^7 + 6*y^2)*x^8 + (9*y^8 + 1)*x^9 + (10*y^9 + 10*y^3)*x^10 + 11*y^10*x^11 + (12*y^11 + 15*y^4 + 4*y)*x^12 + 13*y^12*x^13 + (14*y^13 + 21*y^5)*x^14 + (15*y^14 + 10*y^2)*x^15 + (16*y^15 + 28*y^6 + 1)*x^16 + ...
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PROG
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(PARI) {a(n) = polcoeff( sum(m=1, n, (x^m + 1 +x*O(x^n))^m - 1), n)}
for(n=1, 100, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A338693
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a(n) = Sum_{d|n} d^(d - n/d) * binomial(d, n/d).
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+10
7
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1, 4, 27, 257, 3125, 46665, 823543, 16777312, 387420490, 10000001250, 285311670611, 8916100467712, 302875106592253, 11112006825910963, 437893890380859625, 18446744073716891649, 827240261886336764177, 39346408075296709766628, 1978419655660313589123979
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OFFSET
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1,2
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LINKS
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FORMULA
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G.f.: Sum_{k>=1} ( (k + x^k)^k - k^k ).
If p is prime, a(p) = p^p.
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MATHEMATICA
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a[n_] := DivisorSum[n, #^(# - n/#) * Binomial[#, n/#] &]; Array[a, 20] (* Amiram Eldar, Apr 24 2021 *)
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PROG
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(PARI) a(n) = sumdiv(n, d, d^(d-n/d)* binomial(d, n/d));
(PARI) N=20; x='x+O('x^N); Vec(sum(k=1, N, (k+x^k)^k-k^k))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A318638
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Expansion of Sum_{n>=1} ( (3 + x^n)^n - 3^n ).
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+10
5
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1, 6, 27, 109, 405, 1467, 5103, 17550, 59050, 197100, 649539, 2126991, 6908733, 22325625, 71744625, 229602925, 731794257, 2324602206, 7360989291, 23245524600, 73222475256, 230128853031, 721764371007, 2259440202825, 7060738412026, 22029517662984, 68630377426119, 213516777941712, 663426981193869, 2058911488612863, 6382625094934119, 19765549255048254, 61149666233193318
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = Sum_{d|n} 3^(d - n/d) * binomial(d, n/d). - Seiichi Manyama, Apr 24 2021
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EXAMPLE
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G.f.: A(x) = x + 6*x^2 + 27*x^3 + 109*x^4 + 405*x^5 + 1467*x^6 + 5103*x^7 + 17550*x^8 + 59050*x^9 + 197100*x^10 + 649539*x^11 + 2126991*x^12 + ...
such that
A(x) = x + (3 + x^2)^2 - 3^2 + (3 + x^3)^3 - 3^3 + (3 + x^4)^4 - 3^4 + (3 + x^5)^5 - 3^5 + (3 + x^6)^6 - 3^6 + (3 + x^7)^7 - 3^7 + ...
RELATED SERIES.
The g.f. A(x) equals following series at y = 3:
Sum_{n>=1} ((y + x^n)^n - y^n) = x + 2*y*x^2 + 3*y^2*x^3 + (4*y^3 + 1)*x^4 + 5*y^4*x^5 + (6*y^5 + 3*y)*x^6 + 7*y^6*x^7 + (8*y^7 + 6*y^2)*x^8 + (9*y^8 + 1)*x^9 + (10*y^9 + 10*y^3)*x^10 + 11*y^10*x^11 + (12*y^11 + 15*y^4 + 4*y)*x^12 + 13*y^12*x^13 + (14*y^13 + 21*y^5)*x^14 + (15*y^14 + 10*y^2)*x^15 + (16*y^15 + 28*y^6 + 1)*x^16 + ...
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PROG
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(PARI) {a(n) = polcoeff( sum(m=1, n, (x^m + 3 +x*O(x^n))^m - 3^m), n)}
for(n=1, 100, print1(a(n), ", "))
(PARI) a(n) = sumdiv(n, d, 3^(d-n/d)* binomial(d, n/d)); \\ Seiichi Manyama, Apr 24 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A338694
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a(n) = Sum_{d|n} d^d * binomial(d, n/d).
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+10
4
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1, 8, 81, 1028, 15625, 280017, 5764801, 134219264, 3486784428, 100000031250, 3138428376721, 106993206079936, 3937376385699289, 155568095575106627, 6568408355712921875, 295147905179822588160, 14063084452067724991009, 708235345355351624428356, 37589973457545958193355601
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OFFSET
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1,2
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LINKS
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FORMULA
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G.f.: Sum_{k>=1} ( (k + k * x^k)^k - k^k ) = Sum_{k>=1} k^k * ( (1 + x^k)^k - 1 ).
If p is prime, a(p) = p^(p+1).
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MATHEMATICA
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a[n_] := DivisorSum[n, #^# * Binomial[#, n/#] &]; Array[a, 20] (* Amiram Eldar, Apr 24 2021 *)
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PROG
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(PARI) a(n) = sumdiv(n, d, d^d*binomial(d, n/d));
(PARI) N=20; x='x+O('x^N); Vec(sum(k=1, N, (k+k*x^k)^k-k^k))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A338695
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a(n) = Sum_{d|n} 2^(d-1) * binomial(d, n/d).
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+10
0
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1, 4, 12, 34, 80, 204, 448, 1072, 2308, 5280, 11264, 25088, 53248, 116032, 245920, 527880, 1114112, 2369152, 4980736, 10508880, 22022336, 46193664, 96468992, 201469408, 419430416, 872734720, 1811960832, 3758844096, 7784628224, 16107909312, 33285996544, 68723417856, 141734089728
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graph;
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OFFSET
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1,2
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LINKS
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FORMULA
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G.f.: (1/2) * Sum_{k>=1} ( (2 + 2 * x^k)^k - 2^k ) = Sum_{k>=1} 2^(k-1) * ( (1 + x^k)^k - 1 ).
If p is prime, a(p) = p * 2^(p-1).
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MATHEMATICA
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a[n_] := DivisorSum[n, 2^(# - 1) * Binomial[#, n/#] &]; Array[a, 20] (* Amiram Eldar, Apr 24 2021 *)
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PROG
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(PARI) a(n) = sumdiv(n, d, 2^(d-1)*binomial(d, n/d));
(PARI) N=40; x='x+O('x^N); Vec(sum(k=1, N, (2+2*x^k)^k-2^k)/2)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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