A088815
Expansion of e.g.f. (1-x)^(-1/(1+log(1-x))).
Original entry on oeis.org
1, 1, 4, 24, 190, 1860, 21638, 291158, 4443556, 75779580, 1427272032, 29409572808, 657829667328, 15868725580344, 410543007882408, 11336582934052104, 332736828827893968, 10342443317857993680, 339343476195341474688
Offset: 0
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With[{nn=20},CoefficientList[Series[(1-x)^(-1/(1+Log[1-x])), {x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Nov 29 2011 *)
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x='x+O('x^25); Vec(serlaplace((1-x)^(-1/(1+log(1-x))))) \\ G. C. Greubel, Feb 16 2017
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, sum(k=0, j, k!*abs(stirling(j, k, 1)))*binomial(i-1, j-1)*v[i-j+1])); v; \\ Seiichi Manyama, May 23 2022
A354288
Expansion of e.g.f. (1 + x)^(2/(1 - 2 * log(1+x))).
Original entry on oeis.org
1, 2, 10, 72, 664, 7440, 97712, 1468768, 24825184, 465516672, 9582002688, 214642099584, 5195322070656, 135064965744384, 3752151488840448, 110892824334154752, 3473236656134243328, 114893633354895538176, 4002000861023966189568, 146388324613230926979072
Offset: 0
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With[{nn=20},CoefficientList[Series[(1+x)^(2/(1-2Log[1+x])),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Oct 13 2022 *)
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my(N=20, x='x+O('x^N)); Vec(serlaplace((1+x)^(2/(1-2*log(1+x)))))
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, sum(k=0, j, 2^k*k!*stirling(j, k, 1))*binomial(i-1, j-1)*v[i-j+1])); v;
A354289
Expansion of e.g.f. (1 + x)^(3/(1 - 3 * log(1+x))).
Original entry on oeis.org
1, 3, 24, 276, 4086, 73620, 1557702, 37770138, 1030916484, 31245154164, 1040274476208, 37716394860936, 1478413316987424, 62274364390387656, 2804282634867538248, 134397620584518275928, 6828489621874434752208, 366547074721109281366128
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace((1+x)^(3/(1-3*log(1+x)))))
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, sum(k=0, j, 3^k*k!*stirling(j, k, 1))*binomial(i-1, j-1)*v[i-j+1])); v;
A088820
Numbers k with abundance radius of 8, i.e., abs(sigma(k)-2*k) = 8.
Original entry on oeis.org
22, 56, 130, 184, 368, 836, 1012, 2272, 11096, 17816, 18904, 33664, 45356, 70564, 77744, 85936, 91388, 100804, 128768, 254012, 388076, 391612, 527872, 1090912, 2087936, 2291936, 13174976, 17619844, 29465852, 35021696, 45335936, 120888092, 260378492, 381236216
Offset: 1
22 is in the sequence since sigma(22) = 1 + 2 + 11 + 22 = 36 = 2*22 - 8.
56 is in the sequence since sigma(56) = 1 + 2 + 4 + 7 + 8 + 14 + 28 + 56 = 120 = 2*56 + 8. - _Michael B. Porter_, Jul 20 2016
Cf.
A045770,
A045768,
A055708,
A077374,
A088007,
A088008,
A088010,
A088011,
A088012,
A088818,
A088819.
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[n: n in [1..2*10^7] | Abs(DivisorSigma(1, n) - 2*n) eq 8]; // Vincenzo Librandi, Jul 20 2016
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Select[Range[1, 10^6], Abs[DivisorSigma[1, #] - 2 #] == 8 &] (* Vincenzo Librandi, Jul 20 2016 *)
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is(n)=abs(sigma(n)-2*n)==8 \\ Use, e.g., select(is,[1..10^5]*2). - M. F. Hasler, Jul 19 2016
A256548
Triangle read by rows, T(n,k) = |n,k|*h(k), where |n,k| are the Stirling cycle numbers and h(k) = hypergeom([-k+1,-k],[],1), for n>=0 and 0<=k<=n.
Original entry on oeis.org
1, 0, 1, 0, 1, 3, 0, 2, 9, 13, 0, 6, 33, 78, 73, 0, 24, 150, 455, 730, 501, 0, 120, 822, 2925, 6205, 7515, 4051, 0, 720, 5292, 21112, 53655, 87675, 85071, 37633, 0, 5040, 39204, 170716, 494137, 981960, 1304422, 1053724, 394353
Offset: 0
Triangle starts:
[1]
[0, 1]
[0, 1, 3]
[0, 2, 9, 13]
[0, 6, 33, 78, 73]
[0, 24, 150, 455, 730, 501]
[0, 120, 822, 2925, 6205, 7515, 4051]
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A000262 = lambda n: simplify(hypergeometric([-n+1, -n], [], 1))
A256548 = lambda n,k: A000262(k)*stirling_number1(n,k)
for n in range(7): [A256548(n,k) for k in (0..n)]
Showing 1-5 of 5 results.
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