A100220 -id:A100220 - cp's OEIS frontend

A132325 Decimal expansion of Product_{k>=0} (1+1/10^k).

2, 2, 2, 4, 4, 6, 9, 1, 3, 8, 2, 7, 4, 1, 0, 1, 2, 6, 4, 2, 5, 2, 1, 5, 6, 1, 3, 4, 1, 8, 8, 8, 1, 1, 6, 0, 7, 4, 9, 5, 0, 1, 4, 9, 3, 5, 1, 5, 5, 1, 8, 5, 6, 7, 1, 5, 7, 5, 9, 1, 6, 4, 7, 4, 0, 6, 6, 5, 0, 6, 9, 3, 8, 9, 7, 6, 2, 8, 2, 2, 0, 8, 7, 5, 2, 9, 4, 4, 4, 4, 5, 2, 8, 4, 2, 7, 0, 4, 7, 1, 1, 2, 9, 4, 8

Offset: 1

Hieronymus Fischer, Aug 20 2007

Twice the constant A132326.

			2.22446913827410126425215613418881160749501...

G. C. Greubel, Table of n, a(n) for n = 1..1200
Richard J. McIntosh, Some Asymptotic Formulae for q-Hypergeometric Series, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; alternative link.

Cf. A081845, A067080, A100220, A132019-A132026, A132034-A132038, A132323-A132326, A132271, A132272, A000593.

digits = 105; NProduct[1+1/10^k, {k, 1, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+3] // N[#, digits+3]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
2*N[QPochhammer[-1/10,1/10]] (* G. C. Greubel, Dec 02 2015 *)

prodinf(x=0, 1+(1/10)^x) \\ Altug Alkan, Dec 03 2015

Equals lim sup_{n->oo} Product_{0<=k<=floor(log_10(n))} (1+1/floor(n/10^k)).
Equals lim sup_{n->oo} A132271(n)/n^((1+log_10(n))/2).
Equals lim sup_{n->oo} A132272(n)/n^((log_10(n)-1)/2).
Equals 2*exp(Sum_{n>0} 10^(-n)*Sum_{k|n} -(-1)^k/k) = 2*exp(Sum_{n>0} A000593(n)/(n*10^n)).
Equals lim sup_{n->oo} A132271(n+1)/A132271(n).
Equals 2*(-1/10; 1/10){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 02 2015
Equals sqrt(2) * exp(log(10)/24 + Pi^2/(12*log(10))) * Product_{k>=1} (1 - exp(-2*(2*k-1)*Pi^2/log(10))) (McIntosh, 1995). - Amiram Eldar, May 20 2023

A258458 Number of partitions of n into parts of exactly 3 sorts which are introduced in ascending order.

Original entry on oeis.org

1, 7, 33, 130, 463, 1557, 5031, 15877, 49240, 151116, 460173, 1394645, 4212071, 12693724, 38195286, 114817389, 344911117, 1035659955, 3108817911, 9330152740, 27997803871, 84008165515, 252053831034, 756220333901, 2268778132337, 6806569134920, 20420175154486

Offset: 3

Alois P. Heinz, May 30 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Column k=3 of A256130.

Cf. A320545.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
a:= n-> T(n,3):
seq(a(n), n=3..35);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, k*b[n - i, i, k]]]];
T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i/(i!*(k - i)!), {i, 0, k}];
a[n_] := T[n, 3];
Table[a[n], {n, 3, 35}] (* Jean-François Alcover, May 22 2018, translated from Maple *)

a(n) ~ c * 3^n, where c = 1/(6*Product_{n>=1} (1-1/3^n)) = 1/(6*QPochhammer[1/3, 1/3]) = 1/(6*A100220) = 0.297552056999755698394581... . - Vaclav Kotesovec, Jun 01 2015

A370466 Decimal expansion of Product_{k>=1} 1 / (1 - 1/3^k).

Original entry on oeis.org

1, 7, 8, 5, 3, 1, 2, 3, 4, 1, 9, 9, 8, 5, 3, 4, 1, 9, 0, 3, 6, 7, 4, 8, 6, 2, 9, 6, 0, 1, 3, 7, 0, 3, 5, 3, 5, 7, 1, 8, 7, 9, 6, 0, 0, 8, 2, 0, 7, 9, 3, 9, 6, 1, 4, 2, 7, 4, 3, 0, 0, 4, 8, 2, 2, 8, 8, 4, 8, 1, 8, 1, 1, 6, 0, 9, 6, 9, 2, 7, 3, 2, 3, 7, 4, 0, 5, 5, 6, 9, 3, 7, 1, 1, 0, 3, 9, 4, 3, 8

Offset: 1

Ilya Gutkovskiy, Mar 30 2024

			1.7853123419985341903674862960137035357...

Kelvin Voskuijl, Table of n, a(n) for n = 1..20000

Cf. A000244, A065446, A100220, A132324.

RealDigits[1/QPochhammer[1/3], 10, 120][[1]] (* Vaclav Kotesovec, Mar 31 2024 *)

Equals 1/QPochhammer(1/3). - Vaclav Kotesovec, Mar 31 2024

A065498 Number of invertible n X n matrices mod 6 (i.e., over the ring Z_6).

Original entry on oeis.org

1, 2, 288, 1886976, 489104179200, 4755360379856486400, 1695944421638473850132889600, 21967113634648374162210646578639667200, 10286692771039109536373764545035369981946101760000, 173770439600109774111384717714984362383506603790098046648320000

Offset: 0

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 25 2001

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Jeffrey Overbey, William Traves, and Jerzy Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29, Iss. 1 (2005), pp. 59-72; author's copy.

Column k=6 of A316622.

Cf. A002884, A048651, A053290, A065128, A100220.

a[n_] := 6^(n^2)*Product[(1 - 1/2^k)*(1 - 1/3^k), { k, 1, n} ]; Table[ a[n], {n, 0, 9} ]

a(n) = 6^(n^2) * Product_{k=1..n} ((1 - 1/2^k)(1 - 1/3^k)).
a(n) = A002884(n)*A053290(n). - Geoffrey Critzer, Jan 26 2018
a(n) ~ c * 6^(n^2), where c = A048651 * A100220 = 0.161757743053... . - Amiram Eldar, Jul 06 2025

More terms from Robert G. Wilson v, Nov 28 2001

A371746 Decimal expansion of Product_{k>=0} 1 / (1 + 1/3^k).

Original entry on oeis.org

3, 1, 9, 5, 0, 2, 2, 8, 8, 3, 1, 8, 7, 3, 8, 8, 9, 0, 1, 9, 4, 8, 0, 0, 7, 1, 0, 1, 1, 0, 9, 0, 0, 6, 5, 4, 2, 4, 2, 6, 8, 4, 5, 5, 1, 9, 4, 5, 6, 2, 2, 7, 5, 3, 6, 5, 1, 4, 7, 1, 7, 5, 9, 6, 0, 9, 2, 0, 1, 1, 7, 9, 9, 2, 8, 8, 4, 7, 6, 6, 4, 2, 4, 5, 0, 6, 1, 1, 7, 7, 9, 6, 5, 4, 3, 3, 8, 7, 1, 8

Offset: 0

Ilya Gutkovskiy, Apr 05 2024

			0.31950228831873889019480071011090065424...

Cf. A000244, A083864, A100220, A132323, A370466, A371749, A371752.

RealDigits[1/QPochhammer[-1, 1/3], 10, 100][[1]]

Equals 1 / A132323.

A370467 Decimal expansion of Product_{k>=2} (1 - 1/k^k).

Original entry on oeis.org

7, 1, 9, 1, 5, 4, 5, 0, 0, 9, 6, 5, 0, 1, 0, 2, 4, 6, 6, 5, 4, 4, 6, 9, 3, 1, 0, 4, 1, 4, 5, 7, 8, 3, 9, 5, 3, 8, 6, 1, 1, 2, 9, 3, 8, 0, 3, 1, 6, 6, 5, 4, 9, 1, 5, 7, 7, 0, 3, 5, 6, 4, 9, 9, 1, 8, 3, 4, 8, 4, 6, 4, 0, 6, 5, 4, 1, 2, 1, 3, 3, 5, 8, 7, 2, 9, 4, 9, 5, 4, 1, 0, 2, 5, 9, 4, 7, 5, 7, 9

Offset: 0

Ilya Gutkovskiy, Mar 30 2024

			0.719154500965010246654469310414578395386...

Cf. A000312, A048651, A100220, A370468.

A384368 Number of permutations of [2n] with n inversions.

Original entry on oeis.org

1, 1, 5, 29, 174, 1068, 6655, 41926, 266338, 1703027, 10947079, 70673825, 457927079, 2976282415, 19395654894, 126688273871, 829176461458, 5436687172806, 35703722618623, 234807844921153, 1546217013188447, 10193761267335877, 67275841673522196, 444431529264364506

Offset: 0

Alois P. Heinz, May 27 2025

nonn

			a(0) = 1: the empty permutation.
a(1) = 1: 21.
a(2) = 5: 1342, 1423, 2143, 2314, 3124.
a(3) = 29: 123654, 124563, 124635, 125364, 125436, 126345, 132564, 132645, 134265, 134526, 135246, 142365, 142536, 143256, 152346, 213564, 213645, 214365, 214536, 215346, 231465, 231546, 234156, 241356, 312465, 312546, 314256, 321456, 412356.

Alois P. Heinz, Table of n, a(n) for n = 0..1208

Cf. A008302, A128566.

Cf. A100220.

a:= n-> coeff(series(mul((1-q^j)/(1-q), j=1..2*n), q, n+1), q, n):
seq(a(n), n=0..23);

a(n) = A008302(2n,n).

a(n) ~ c * 3^(3*n - 1/2) / (sqrt(Pi*n) * 2^(2*n)), where c = QPochhammer(1/3) = A100220 = 0.5601260779279489449697922433141400143797363337983... - Vaclav Kotesovec, Jun 09 2025

A069920 Number of noninvertible n X n matrices mod 6 (i.e., over the ring Z_6).

Original entry on oeis.org

4, 1008, 8190720, 2332005728256, 23674927650073214976, 8618480376852061696039059456, 112746432609478969278312620164116381696, 53053593891934168169788522401776516627950360068096, 898369021875992553077210146021642426986208034219512412195782656

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 05 2002

nonn

Cf. A048651, A065498, A069580, A100220 .

a[n_] := 6^(n^2) * (1 - Product[(1 - 1/2^k) * (1 - 1/3^k), {k, 1, n}]); Array[a, 10] (* Amiram Eldar, Jul 12 2025 *)

a(n) = 6^(n^2)*(1-prod(k=1,n,(1-1/2^k)*(1-1/3^k)));

a(n) = 6^(n^2) - A065498(n).

a(n) ~ c * 6^(n^2), where c = 1 - A048651 * A100220 = 0.838242256946... . - Amiram Eldar, Jul 12 2025

More terms from Benoit Cloitre, May 11 2002

cp's OEIS Frontend

A132325 Decimal expansion of Product_{k>=0} (1+1/10^k).

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A258458 Number of partitions of n into parts of exactly 3 sorts which are introduced in ascending order.

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A370466 Decimal expansion of Product_{k>=1} 1 / (1 - 1/3^k).

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A065498 Number of invertible n X n matrices mod 6 (i.e., over the ring Z_6).

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A371746 Decimal expansion of Product_{k>=0} 1 / (1 + 1/3^k).

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A370467 Decimal expansion of Product_{k>=2} (1 - 1/k^k).

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A384368 Number of permutations of [2n] with n inversions.

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A069920 Number of noninvertible n X n matrices mod 6 (i.e., over the ring Z_6).

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