A350366 -id:A350366 - cp's OEIS frontend

A350376 a(n) = [x^n] Product_{k=1..n} 1/(1 - k*x)^2.

1, 2, 23, 480, 14627, 587580, 29331038, 1750923328, 121673580435, 9648709656300, 859874920598850, 85078769750118144, 9254316901029412110, 1097635452798476278232, 140986468651523106196060, 19496446561112852736019200, 2887977880849714395963280515

Offset: 0

Seiichi Manyama, Dec 27 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Cf. A007820, A008277, A129256, A298851, A350366, A383862, A384060.

a[n_] := Coefficient[Series[Product[1/(1 - k*x)^2, {k, 1, n}], {x, 0, n}], x, n]; Array[a, 17, 0] (* Amiram Eldar, Dec 28 2021 *)
Table[Sum[StirlingS2[n + k, n]*StirlingS2[2*n - k, n], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 29 2021 *)

a(n) = sum(k=0, n, stirling(n+k, n, 2)*stirling(2*n-k, n, 2));

a(n) = Sum_{k=0..n} Stirling2(n+k, n) * Stirling2(2*n-k, n).

a(n) ~ c * d^n * (n-1)!, where d = 27 / (4*LambertW(-3*exp(-3/2)/2)^2 * (3 + 2*LambertW(-3*exp(-3/2)/2))) = 9.858422414446789720857925020919293523149... and c = sqrt(3/(-LambertW(-3*exp(-3/2)/2) * (1 + LambertW(-3*exp(-3/2)/2)))) / (4*Pi) = 0.28482428628793763109169664913715827647091747... - Vaclav Kotesovec, Dec 28 2021, updated May 14 2025

A351764 a(n) = [x^n] Product_{k=1..n} (1 + kx)^n / (1 - kx)^n.

Original entry on oeis.org

1, 2, 72, 7848, 1728000, 641258850, 360403076376, 285818177146208, 304172586944446464, 418400927094822149970, 722587619114932445325000, 1530927286486636135080191736, 3904621941927926455303092180480, 11801667653769333351640692783069714

Offset: 0

Vaclav Kotesovec, Feb 18 2022

nonn

Cf. A350366, A351507, A351508.

Table[SeriesCoefficient[Product[(1 + k*x)^n / (1 - k*x)^n, {k,1,n}], {x,0,n}], {n,0,20}]

a(n) = my(x='x+O(x^(n+1))); polcoef(prod(k=1, n, (1+k*x)^n/ (1-k*x)^n), n); \\ Michel Marcus, Feb 19 2022

a(n) ~ exp(n + 1) * n^(2*n - 1/2) / sqrt(2*Pi).

A338328 G.f. A(x) satisfies: [x^n] (1 + nx - A(x))^(2n-1) = 0, for n > 0.

Original entry on oeis.org

1, 1, 8, 129, 3216, 108770, 4638624, 238318885, 14304161568, 981167968494, 75656236536880, 6474624435825546, 608726166485138400, 62351661805733365988, 6910034200942823999216, 823702177681649409615885, 105083183676401369775220288, 14284797856254053619382213974

Offset: 1

Paul D. Hanna, Oct 24 2020

nonn

Compare to: [x^n] (1 + n*x - C(x))^(n+1) = 0, for n>0, where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

Compare to: [x^n] (1 + n*x - W(x))^n = 0, for n>0, where W(x) = Sum_{n>=1} (n-1)^(n-1)*x^n/n! = 1 + x/LambertW(-x).

			G.f.: A(x) = x + x^2 + 8*x^3 + 129*x^4 + 3216*x^5 + 108770*x^6 + 4638624*x^7 + 238318885*x^8 + 14304161568*x^9 + 981167968494*x^10 + 75656236536880*x^11 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in (1 + n*x - A(x))^(2*n-1) begins:
n=0: [1, 1, 2, 11, 150, 3522, 116004, 4875971, 248249926, ...];
n=1: [1, 0, -1, -8, -129, -3216, -108770, -4638624, -238318885, ...];
n=2: [1, 3, 0, -29, -435, -10395, -344980, -14551710, -742232889, ...];
n=3: [1, 10, 35, 0, -995, -22108, -685040, -27845120, -1387317400, ...];
n=4: [1, 21, 182, 763, 0, -44352, -1308496, -49463691, -2359476623, ...];
n=5: [1, 36, 567, 5016, 24795, 0, -2460798, -90517248, -4024228311, ...];
n=6: [1, 55, 1364, 19987, 188111, 1076779, 0, -163739444, -7210725280, ...];
n=7: [1, 78, 2795, 60736, 886665, 8986692, 58632756, 0, -12708845682, ...];
n=8: [1, 105, 5130, 154475, 3196890, 47836926, 523400300, 3840855735, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1 + n*x - A(x))^(2*n-1) = 0, for n > 0.
ODD TERMS.
The odd terms seem to occur only at positions equal to powers of 2: a(1), a(2), a(4), a(8), a(16), ...; the odd terms begin: [1, 1, 129, 238318885, 823702177681649409615885, 1728309826852815676925583353579702599032084743126745454358749, ...].
RELATED SERIES.
1/(1 - A(x)) = 1 + x + 2*x^2 + 11*x^3 + 150*x^4 + 3522*x^5 + 116004*x^6 + 4875971*x^7 + 248249926*x^8 + 14807944838*x^9 + 1011137601996*x^10 + ...
Series_Reversion(A(x)) = x - x^2 - 6*x^3 - 94*x^4 - 2404*x^5 - 83808*x^6 - 3667808*x^7 - 192327976*x^8 - 11726343040*x^9 - 814059155216*x^10 - ...

Paul D. Hanna, Table of n, a(n) for n = 1..300

Cf. A337758, A000108, A018900, A350366.

{a(n) = my(A=[1],m=1); for(i=1,n, A=concat(A,0);
m=#A; A[#A] = polcoeff( (1 + m*x - x*Ser(A))^(2*m-1), m)/(2*m-1) );A[n]}
for(n=1,30,print1(a(n),", "))

a(n) is odd iff n is a power of 2 (conjecture).
a(n) = 2 (mod 4) iff n is twice the sum of two distinct powers of 2 (conjecture).
a(n) ~ c * d^n * n! / n^2, where d = (1+r) / ((-1 + exp(r + LambertW(-1, -exp(-r)*r))) * LambertW(-exp(-1-r)*(1+r))) = 8.406107401279769476199925123910168..., r = 0.7545302104650497245839827141610818561001159135034... is the root of the equation r*(1 + r + LambertW(-exp(-1 - r)*(1 + r))) = -(1 + r)*(r + LambertW(-1, -exp(-r)*r)) and c = 0.0183535737... - Vaclav Kotesovec, Oct 25 2020, updated Dec 29 2021

A383900 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of Product_{j=0..k} (1 + jx)/(1 - jx).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 6, 2, 0, 1, 12, 18, 2, 0, 1, 20, 72, 42, 2, 0, 1, 30, 200, 312, 90, 2, 0, 1, 42, 450, 1400, 1152, 186, 2, 0, 1, 56, 882, 4650, 8000, 3912, 378, 2, 0, 1, 72, 1568, 12642, 38250, 40520, 12672, 762, 2, 0, 1, 90, 2592, 29792, 142002, 271770, 190400, 39912, 1530, 2, 0

Offset: 0

Seiichi Manyama, May 14 2025

			Square array begins:
  1, 1,   1,    1,     1,      1, ...
  0, 2,   6,   12,    20,     30, ...
  0, 2,  18,   72,   200,    450, ...
  0, 2,  42,  312,  1400,   4650, ...
  0, 2,  90, 1152,  8000,  38250, ...
  0, 2, 186, 3912, 40520, 271770, ...

Columns k=0..4 give A000007, A040000, A068293(n+1), A383910, A383911.
Main diagonal gives A350366.
A(n,n-1) gives A383767.
Cf. A383818, A383843.

a(n, k) = sum(j=0, k, abs(stirling(k+1, j+1, 1))*stirling(j+n, k, 2));

A(n,k) = Sum_{j=0..k} |Stirling1(k+1,j+1)| * Stirling2(j+n,k).

A384043 a(n) = [x^n] Product_{k=1..n} (1 + k^2x) / (1 - k^2x).

Original entry on oeis.org

1, 2, 50, 4188, 735600, 221302710, 101667388082, 66218673102680, 58048466179356672, 65901249246347377770, 94061755750395244537250, 164863945136411230998746612, 348110204753572939058548570000, 871547135491620353615820806025918, 2552918049709989779004770502542335650

Offset: 0

Vaclav Kotesovec, May 18 2025

nonn

Cf. A001044, A298851.

Cf. A350366, A384044, A351764.

Table[SeriesCoefficient[Product[(1+k^2*x)/(1-k^2*x), {k, 1, n}], {x, 0, n}], {n, 0, 15}]

a(n) ~ c * d^n * n!^2 / n^(3/2), where d = 16.6871576653578743696262746377576281620174969944584774545888... and c = 0.1371163625236187865398447973928851799479072107076663329994...

A338408 E.g.f. A(x) satisfies: [x^n] (1 + nx - A(x))^(2n) = 0, for n > 0.

Original entry on oeis.org

1, 3, 70, 4515, 567576, 116389295, 35111089728, 14574226069095, 7944376570503040, 5494208894263886139, 4694820247236686649600, 4853712224007783889422923, 5968210130160831707746406400, 8605241830169634366425696447655, 14375558607944255605507888571539456

Offset: 1

Paul D. Hanna, Oct 24 2020

nonn

Compare to: [x^n] (1 + n*x - W(x))^n = 0, for n>0, where W(x) = Sum_{n>=1} (n-1)^(n-1)*x^n/n! = 1 + x/LambertW(-x).

Compare to: [x^n] (1 + n*x - C(x))^(n+1) = 0, for n>0, where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

			E.g.f.: A(x) = x + 3*x^2/2! + 70*x^3/3! + 4515*x^4/4! + 567576*x^5/5! + 116389295*x^6/6! + 35111089728*x^7/7! + 14574226069095*x^8/8! + 7944376570503040*x^9/9! + 5494208894263886139*x^10/10! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in (1 + n*x - A(x))^(2*n) begins:
n=0: [1, 0, 0, 0, 0, 0, 0, 0, ...];
n=1: [1, 0, -6, -140, -8976, -1130952, -232274240, -70128541380, ...];
n=2: [1, 4, 0, -364, -21504, -2530284, -504753152, -149907313980, ...];
n=3: [1, 12, 102, 0, -45960, -5063916, -928551600, -263868802728, ...];
n=4: [1, 24, 480, 7000, 0, -9924168, -1748523008, -457324971720, ...];
n=5: [1, 40, 1410, 42140, 939360, 0, -3259331360, -836926230780, ...];
n=6: [1, 60, 3264, 158220, 6595584, 208807788, 0, -1509806731620, ...];
n=7: [1, 84, 6510, 460936, 29355816, 1626947196, 69489455728, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1 + n*x - A(x))^(2*n) = 0, for n > 0.

Paul D. Hanna, Table of n, a(n) for n = 1..200

Cf. A338328, A337758, A350366.

{a(n) = my(A=[1],m=1); for(i=1,n, A=concat(A,0); m=#A; A[#A] = polcoeff( (1 + m*x - x*Ser(A))^(2*m), m)/(2*m) ); n!*A[n]}
for(n=1,30,print1(a(n),", "))

a(n) ~ c * d^n * n!^2 / n^2, where d = (1+r) / ((-1 + exp(r + LambertW(-1, -exp(-r)*r))) * LambertW(-exp(-1-r)*(1+r))) = 8.406107401279769476199925123910168..., r = 0.7545302104650497245839827141610818561001159135034... is the root of the equation r*(1 + r + LambertW(-exp(-1 - r)*(1 + r))) = -(1 + r)*(r + LambertW(-1, -exp(-r)*r)) and c = 0.031468237083... - Vaclav Kotesovec, Aug 12 2021, updated Dec 29 2021

A383767 a(n) = [x^n] Product_{k=0..n-1} (1 + kx)/(1 - kx).

Original entry on oeis.org

1, 0, 2, 42, 1152, 40520, 1751850, 90087522, 5376546560, 365487900192, 27886922161650, 2360357986720250, 219495753481590432, 22246783602163580616, 2440974108105319141082, 288270640787372104920450, 36459004369727317927680000, 4916744437454382604092493952, 704282170015570676249171941218

Offset: 0

Seiichi Manyama, May 14 2025

nonn

Cf. A350366.

a(n) = polcoef(prod(k=0, n-1, (1+k*x)/(1-k*x)+x*O(x^n)), n);

a(n) = Sum_{k=0..n} |Stirling1(n,k)| * Stirling2(k+n-1,n-1) for n > 0.

A384044 a(n) = [x^n] Product_{k=1..n} (1 + k^3x) / (1 - k^3x).

Original entry on oeis.org

1, 2, 162, 75672, 104312000, 317309605650, 1803288012589602, 17180843554017736544, 254292459616733559570432, 5525508321588276184345621650, 168733575675064578625834983478850, 6994229599670887851052241626545021912, 382562895157136117988572795915676719695680

Offset: 0

Vaclav Kotesovec, May 18 2025

nonn

Cf. A000442, A351800.

Cf. A350366, A384043, A351764.

Table[SeriesCoefficient[Product[(1+k^3*x)/(1-k^3*x), {k, 1, n}], {x, 0, n}], {n, 0, 12}]

a(n) ~ c * d^n * n!^3 / n^2, where d = 37.604795475701444958019770120055586495991039059348094619704... and c = 0.063895861310548119570865800164582089372152350471371583403...

A384086 a(n) = [x^n] Product_{k=1..n} ((1 + kx)/(1 - kx))^2.

Original entry on oeis.org

1, 4, 72, 2352, 112000, 7023540, 546991704, 50923706176, 5517464159232, 682067031126660, 94744306830613000, 14610279918692775504, 2476682373835289303424, 457771369968515293229812, 91624876032673265663215800, 19743379886572250897986694400, 4556982707091255612929249419264

Offset: 0

Vaclav Kotesovec, May 19 2025

nonn

Cf. A129256, A350376.

Cf. A350366, A384087, A384088, A351764.

Table[SeriesCoefficient[Product[(1+k*x)^2/(1-k*x)^2, {k, 1, n}], {x, 0, n}], {n, 0, 20}]

a(n) ~ c * d^n * n! / n, where d = 15.357995623209995052090556511543938190953157405669200... and c = 0.3746298100044008083790505105262276548713201624206421...

A384087 a(n) = [x^n] Product_{k=1..n} ((1 + kx)/(1 - kx))^3.

Original entry on oeis.org

1, 6, 162, 7848, 552000, 51035310, 5853933666, 802178739936, 127879052859648, 23252775004089990, 4750089647035004250, 1077069265550569663416, 268437124701985949614944, 72940650531961450912140558, 21461129870889481564510048050, 6797577340761206051865208521600, 2306127185536355501260494657447936

Offset: 0

Vaclav Kotesovec, May 19 2025

nonn

Cf. A384012, A383862.

Cf. A350366, A384086, A384088, A351764.

Table[SeriesCoefficient[Product[(1+k*x)^3/(1-k*x)^3, {k, 1, n}], {x, 0, n}], {n, 0, 16}]

a(n) ~ c * d^n * n! / n, where d = 22.56625698335414867480351407039325848948214595770919713967057... and c = 0.403760467212667768540403611728406212428403946576093482938996...

cp's OEIS Frontend

A350376 a(n) = [x^n] Product_{k=1..n} 1/(1 - k*x)^2.

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A351764 a(n) = [x^n] Product_{k=1..n} (1 + k*x)^n / (1 - k*x)^n.

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A338328 G.f. A(x) satisfies: [x^n] (1 + n*x - A(x))^(2*n-1) = 0, for n > 0.

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A383900 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of Product_{j=0..k} (1 + j*x)/(1 - j*x).

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A384043 a(n) = [x^n] Product_{k=1..n} (1 + k^2*x) / (1 - k^2*x).

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A338408 E.g.f. A(x) satisfies: [x^n] (1 + n*x - A(x))^(2*n) = 0, for n > 0.

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A383767 a(n) = [x^n] Product_{k=0..n-1} (1 + k*x)/(1 - k*x).

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A384044 a(n) = [x^n] Product_{k=1..n} (1 + k^3*x) / (1 - k^3*x).

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A384086 a(n) = [x^n] Product_{k=1..n} ((1 + k*x)/(1 - k*x))^2.

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A351764 a(n) = [x^n] Product_{k=1..n} (1 + kx)^n / (1 - kx)^n.

A338328 G.f. A(x) satisfies: [x^n] (1 + nx - A(x))^(2n-1) = 0, for n > 0.

A383900 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of Product_{j=0..k} (1 + jx)/(1 - jx).

A384043 a(n) = [x^n] Product_{k=1..n} (1 + k^2x) / (1 - k^2x).

A338408 E.g.f. A(x) satisfies: [x^n] (1 + nx - A(x))^(2n) = 0, for n > 0.

A383767 a(n) = [x^n] Product_{k=0..n-1} (1 + kx)/(1 - kx).

A384044 a(n) = [x^n] Product_{k=1..n} (1 + k^3x) / (1 - k^3x).

A384086 a(n) = [x^n] Product_{k=1..n} ((1 + kx)/(1 - kx))^2.

A384087 a(n) = [x^n] Product_{k=1..n} ((1 + kx)/(1 - kx))^3.