A007047 -id:A007047 - cp's OEIS frontend

A260464 Number of chains in the poset of even-sized subsets of {1,2,...,n} ordered by inclusion.

1, 1, 3, 7, 27, 91, 483, 2227, 15627, 92491, 810963, 5866147, 61720827, 527635291, 6476783043, 63886537267, 896245131627, 10019232896491, 158126425788723, 1975680877259587, 34645295464104027, 478434297205284091, 9228741116739540003, 139581878985127217107

Offset: 0

Geoffrey Critzer, Jul 26 2015

nonn

			a(3)=7 because there are 4 chains of length zero: {{}}; {{1,2}}; {{1,3}}; {{2,3}} and there are 3 chains of length one: {{},{1,2}}; {{},{1,3}}; {{},{2,3}}.

Cf. A007047, A260504.

nn = 20; c = Cosh[x] - 1; s = Sinh[x];Range[0, nn]! CoefficientList[Series[(c^2 + 2 c + 1 + s c + s)/(1 - c), {x, 0, nn}], x]

my(x='x+O('x^33), c = cosh(x)-1, s=sinh(x)); Vec(serlaplace( (c^2 + 2*c + 1 + s*c + s)/(1 - c) )) \\ Joerg Arndt, Jul 27 2015

E.g.f.: (c^2 + 2*c + 1 + s*c + s)/(1 - c) where c = cosh(x)-1 and s=sinh(x).

a(n) ~ n! * (4/sqrt(3)+2 + (4/sqrt(3)-2)*(-1)^n) / log(2+sqrt(3))^(n+1). - Vaclav Kotesovec, Jul 27 2015

A260504 Number of chains in the poset of all odd-sized subsets of {1,2,...,n} ordered by inclusion.

Original entry on oeis.org

0, 1, 2, 7, 20, 91, 362, 2227, 11720, 92491, 608222, 5866147, 46290620, 527635291, 4857587282, 63886537267, 672183848720, 10019232896491, 118594819341542, 1975680877259587, 25983971598078020, 478434297205284091, 6921555837554655002, 139581878985127217107

Offset: 0

Geoffrey Critzer, Jul 27 2015

nonn

			a(4) = 20 because there are C(4,1) + C(4,3) = 8 chains of length zero (these are the odd-sized subsets of {1,2,3,4}). There are 12 chains of length one: {{1},{1,2,3}}; {{1},{1,2,4}}; {{1},{1,3,4}}; {{2},{1,2,3}}; {{2},{1,2,4}}; {{2},{2,3,4}}; {{3},{1,2,3}}; {{3},{1,3,4}}; {{3},{2,3,4}}; {{4},{1,2,4}}; {{4},{1,3,4}}; {{4},{2,3,4}}.

Cf. A260464, A007047.

# Assuming a(0) = 1:
p := proc(n, z) option remember; local k; if n = 0 then 1 else
normal(add(`if`(k mod 2 = 1, 0, binomial(n, k)*p(k, 0)*(z+1)^(n-k-1)), k=0..n-1))
fi end: A260504 := n -> p(n, 1): seq(A260504(n), n = 0..23); # Peter Luschny, Jun 19 2023

nn = 20; c=Cosh[x]-1;s=Sinh[x];Range[0,nn]!CoefficientList[Series[(s^2 + s c + s)/(1 - c), {x, 0, nn}], x]

E.g.f.: (s^2 + s*c + s)/(1 - c) where s = sinh(x) and c = cosh(x) - 1.

a(n) ~ n! * (sqrt(3)+2 + (-1)^n*(sqrt(3)-2)) / log(2+sqrt(3))^(n+1). - Vaclav Kotesovec, Jul 27 2015

A292915 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)/(2 - exp(x)).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 13, 1, 4, 11, 26, 75, 1, 5, 18, 51, 150, 541, 1, 6, 27, 94, 299, 1082, 4683, 1, 7, 38, 161, 582, 2163, 9366, 47293, 1, 8, 51, 258, 1083, 4294, 18731, 94586, 545835, 1, 9, 66, 391, 1910, 8345, 37398, 189171, 1091670, 7087261, 1, 10, 83, 566, 3195, 15666, 74067, 378214, 2183339, 14174522, 102247563

Offset: 0

Ilya Gutkovskiy, Sep 26 2017

A(n,k) is the k-th binomial transform of A000670 evaluated at n.

			E.g.f. of column k: A_k(x) = 1 + (k + 1)*x/1! + (k^2 + 2*k + 3)*x^2/2! + (k^3 + 3*k^2 + 9*k + 13)*x^3/3! +  (k^4 + 4*k^3 + 18*k^2 + 52*k + 75) x^4/4! + ...
Square array begins:
    1,     1,     1,     1,     1,      1,  ...
    1,     2,     3,     4,     5,      6,  ...
    3,     6,    11,    18,    27,     38,  ...
   13,    26,    51,    94,   161,    258,  ...
   75,   150,   299,   582,  1083,   1910,  ...
  541,  1082,  2163,  4294,  8345,  15666,  ...

G. C. Greubel, Antidiagonals n = 0..50, flattened
N. J. A. Sloane, Transforms

Columns k=0..4 give A000670, A000629, A007047, A259533, A368317.
Rows n=0..2 give A000012, A000027, A102305.
Main diagonal gives A292916.

R:=PowerSeriesRing(Rationals(), 50);
T:= func< n,k | Coefficient(R!(Laplace( Exp(k*x)/(2-Exp(x)) )), n) >;
A292915:= func< n,k | T(k,n-k) >;
[A292915(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 12 2024

A:= proc(n, k) option remember; k^n +add(
       binomial(n, j)*A(j, k), j=0..n-1)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 27 2017

Table[Function[k, n! SeriesCoefficient[Exp[k x]/(2 - Exp[x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, HurwitzLerchPhi[1/2, -n, k]/2][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

a000670(n) = sum(k=0, n, k!*stirling(n, k, 2));
A(n, k) = 2^k*a000670(n)-sum(j=0, k-1, 2^j*(k-1-j)^n); \\ Seiichi Manyama, Dec 25 2023

def T(n,k): return factorial(n)*( exp(k*x)/(2-exp(x)) ).series(x, n+1).list()[n]
def A292915(n,k): return T(k,n-k)
flatten([[A292915(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jun 12 2024

E.g.f. of column k: exp(k*x)/(2 - exp(x)).

A(n,k) = 2^k*A000670(n) - Sum_{j=0..k-1} 2^j*(k-1-j)^n. - Seiichi Manyama, Dec 25 2023

A107954 Number of chains in the power set lattice, or the number of fuzzy subsets of an (n+4)-element set X_(n+4) with specification n elements of one kind, 3 elements of another and 1 of yet another kind.

Original entry on oeis.org

79, 527, 2415, 9263, 31871, 101759, 307455, 890111, 2490367, 6774783, 18001919, 46886911, 120029183, 302678015, 753205247, 1852375039, 4507828223, 10866393087, 25970081791, 61583917055, 144997089279, 339159810047

Offset: 0

Venkat Murali (v.murali(AT)ru.ac.za), May 30 2005

This sequence is an example of another line in a triple sequence A(n,m,l) with n a nonnegative integer, m = 2 and l = 1. It is related to sequences A107464, A107953 which are part of the same triple sequence with different parameter values for m and l.

			a(2) = 8 * ( (16 + 184)/6 + (316 + 370)/3 + 40 ) - 1 = 2415. This is the number of fuzzy subsets of a set of (2+4) elements of which 2 are of one kind, 3 are of another kind and 1 of a kind distinct from the other two.

V. Murali, On the enumeration of fuzzy subsets of X_(n+4) of specification n^1 3^1 1, Rhodes University JRC-Abstract-Report, In Preparation, 12 pages 2005.

Venkat Murali, Home page.
Index entries for linear recurrences with constant coefficients, signature (11,-50,120,-160,112,-32).

Cf. A007047, A107392, A107464, A107953.

a[n_] := 2^n(n^4 + 23n^3 + 158n^2 + 370n + 240)/3 - 1; Table[ a[n], {n, 0, 21}] (* Robert G. Wilson v, May 31 2005 *)
LinearRecurrence[{11,-50,120,-160,112,-32},{79,527,2415,9263,31871,101759},40] (* Harvey P. Dale, Aug 15 2025 *)

a(n) = 2^(n+1)*( (n^4 + 23*n^3)/6 + (79*n^2 + 185*n)/3 + 40 ) - 1.

G.f.: (128*x^4-432*x^3+568*x^2-342*x+79) / ((x-1)*(2*x-1)^5). [Colin Barker, Dec 10 2012]

a(6)-a(21) from Robert G. Wilson v, May 31 2005

A330032 The number of chains of strictly rooted upper triangular or lower triangular matrices of order n.

Original entry on oeis.org

1, 2, 26, 9366, 204495126, 460566381955706, 162249649997008147763642, 12595124129900132067036747870669270, 288398561903310939256721956218813835167026180310, 2510964964470962082968627390938311899485883615067802615950711482

Offset: 0

S. R. Kannan, Rajesh Kumar Mohapatra, Feb 29 2020

nonn

Also, the number of chains in the power set of (n^2-n)/2-elements such that the first term of the chains is either an empty set or a set of (n^2-n)/2-elements.
The number of rooted chains of 2-element subsets of {0,1, 2, ..., n} that contain no consecutive integers.
The number of distinct rooted reflexive symmetric fuzzy matrices of order n.
The number of chains in the set consisting of all n X n reflexive symmetric matrices such that the first term of the chains is either reflexive symmetric matrix or unit matrix.

Alois P. Heinz, Table of n, a(n) for n = 0..28
S. R. Kannan and Rajesh Kumar Mohapatra, Counting the Number of Non-Equivalent Classes of Fuzzy Matrices Using Combinatorial Techniques, arXiv preprint arXiv:1909.13678 [math.GM], 2019.
R. B. Nelsen and H. Schmidt, Jr., Chains in power sets, Math. Mag., 64 (1) (1991), 23-31.
M. Tărnăuceanu, The number of chains of subgroups of a finite elementary abelian p-group, arXiv preprint arXiv:1506.08298 [math.GR], 2015.

Cf. A000629, A038719, A007047, A328044, A330301, A330302, A330804, A331957.

a(n) = A000629((n^2-n)/2).

Missing term a(6) = 162249649997008147763642 inserted by Georg Fischer, Jul 15 2024

A365863 a(0) = 1; thereafter a(n) = nSum_{k = 0..n-1} binomial(n, k)(-1)^(1+n+k)*a(k).

Original entry on oeis.org

1, 1, 2, 12, 156, 3380, 108930, 4876242, 289111032, 21916777752, 2067208751790, 237380181141950, 32601704893973556, 5276471519805880836, 993835167745129599162, 215520207875112312124890, 53311353846240820033325040, 14919977169758349265112350256, 4690364757880376663319746737926

Offset: 0

Thomas Scheuerle, Nov 09 2023

nonn

Let P_k(x) be the polynomial of order k which satisfies a(m) = P_k(m) for m = 0..k, then a(k+1) = k * P_k(k+1).
This sequence is a member of a family of sequences with related properties. Here are some examples:
With b(k+1) = 1 + P_k(k+1) we get b(k) = A000079(k).
With b(k+1) = 2 + P_k(k+1) we get b(k) = A000225(k).
With b(k+1) = 3 + P_k(k+1) we get b(k) = A033484(k).
With b(k+1) = 2 * P_k(k+1) we get b(k) = A000629(k).
With b(k+1) = 1 + 2 * P_k(k+1) we get b(k) = A007047(k).
With b(k+1) = 3 * P_k(k+1) we get b(k) = A201339(k).
With b(k+1) = 5 * P_k(k+1) we get b(k) = A201365(k).
With b(k+1) = -1 * P_k(k+1) we get b(k) = A000670(k)*(-1)^k.
With b(k+1) = -2 * P_k(k+1) we get b(k) = A004123(k+1)*(-1)^k.
With b(k+1) = -3 * P_k(k+1) we get b(k) = A032033(k)*(-1)^k.
With b(k+1) = -4 * P_k(k+1) we get b(k) = A094417(k)*(-1)^k.
With b(k+1) = -m * P_k(k+1) we get b(k) = Bo(m, k)*(-1)^k, Bo(m, k) are Generalized ordered Bell numbers.

Cf. A000079, A000225, A000629, A000670.
Cf. A004123, A007047, A032033, A033484.
Cf. A094417, A201339, A201365, A367308.

a[n_] := a[n] = If[n == 0, 1, n*Sum[Binomial[n, k]*(-1)^(1 + n + k)*a[k], {k, 0, n - 1}]]; Table[a[n], {n, 0, 20}] (* Vaclav Kotesovec, Nov 12 2023 *)

a(n) = if(n == 0, 1,sum(k = 0,n-1, n*binomial(n, k)*(-1)^(1+n+k)*a(k)))

a(n) ~ c * n^(2*n + 1/2) / exp(2*n), where c = 2.9711739498821842863440481701659942323709511474486414... - Vaclav Kotesovec, Nov 12 2023

A368317 Expansion of e.g.f. exp(4*x) / (2 - exp(x)).

Original entry on oeis.org

1, 5, 27, 161, 1083, 8345, 74067, 754241, 8726283, 113375465, 1635899907, 25961939921, 449464541883, 8429731963385, 170261482711347, 3684531041231201, 85050474868523883, 2085932272336772105, 54168554611721580387, 1484825397108091268081

Offset: 0

Seiichi Manyama, Dec 21 2023

nonn

Cf. A000670, A000629, A007047, A259533.

b(n, t) = sum(k=0, n, t^k*k!*stirling(n, k, 2));
a(n, m=4, t=1) = my(u=1+1/t); u^m*b(n, t)-(1/t)*sum(j=0, m-1, u^j*(m-1-j)^n);

a(n) = 4^n + Sum_{k=1..n} binomial(n,k) * a(n-k).

a(n) = 16*A000670(n) - (3^n + 2*2^n + 4 + 8*0^n).

A107955 Number of chains in the power set lattice or the number of fuzzy subsets of an (n+5)-element set X_(n+5) with specification n elements of one kind, 4 elements of another and 1 of yet another kind.

Original entry on oeis.org

191, 1471, 7551, 31871, 119231, 410303, 1327103, 4090623, 12130303, 34842623, 97435647, 266313727, 713637887, 1879523327, 4875091967, 12474187775, 31531728895, 78832992255, 195135799295, 478649778175, 1164351373311

Offset: 0

Venkat Murali (v.murali(AT)ru.ac.za), Jun 01 2005

This sequence is another example, together with A107953 and A107954, of a triple sequence A(n,m,l) with n a nonnegative integer, m = 4 and l = 1.

			a(3) = (2^(3+1))*(1/24)*(3^5 + 36 * 3^4 + 431 * 3^3 + 2088 * 3^2 + 3972 * 3 + 2304) - 1 = 31871. This is the number of chains in the power set lattice (which is also the number of fuzzy subsets) of X_(n+5).

Venkat Murali, On the enumeration of fuzzy subsets of an (n+5)-element set X_(n+5) of specification n^1 4^1 1, Rhodes University JRC-Abstract-Report, In Preparation, 15 pages 2005.

Venkat Murali, Home page.
Index entries for linear recurrences with constant coefficients, signature (13,-72,220,-400,432,-256,64).

Cf. A007047, A107392, A107464, A107953, A107954.

a(n) = (2^(n+1))*(1/24)*(n^5 + 36*n^4 + 431*n^3 + 2088*n^2 + 3972*n + 2304) - 1,

G.f.: (320*x^5-1360*x^4+2400*x^3-2180*x^2+1012*x-191) / ((x-1)*(2*x-1)^6). [Colin Barker, Dec 10 2012]

A299404 a(n) = 1 + Sum_{m >= 1} (m + 1)^n/2^(m - 1).

Original entry on oeis.org

3, 7, 23, 103, 599, 4327, 37463, 378343, 4366679, 56698087, 817980503, 12981060583, 224732540759, 4214866787047, 85130743763543, 1842265527822823, 42525237455850839, 1042966136233087207, 27084277306054762583, 742412698554627289063, 21421502369955073624919

Offset: 0

Joseph Wheat, Feb 20 2018

nonn

Cf. A007047, A083652, A162509.

Table[1 + LerchPhi[1/2, -n, 2], {n, 0, 20}] (* Vaclav Kotesovec, Apr 17 2018 *)

a(n) = 1+ round(suminf(m=1, (m + 1)^n/2^(m - 1)));

a(n + 1) = 4*A162509(n + 1) + a(n).
a(n) = 2*A007047(n) + 1.
{a(4n - 3), a(4n - 2), a(4n - 1), a(4n)} mod 10 = {7, 3, 3, 9} for n > 0.
floor(log_2(a(n))) = A083652(n).
Lim_{n->infinity} (a(n)^(1/n))/n = 1/(e*log(2)). - Jon E. Schoenfield, Feb 24 2018
a(n)/n! ~ 4 / (log(2))^(n+1). - Vaclav Kotesovec, Apr 17 2018

A329712 The number of rooted chains in the lattice of (0, 1) matrices of order n.

Original entry on oeis.org

1, 2, 150, 14174522, 10631309363962710, 213394730876951551651166996282, 288398561903310939256721956218813835167026180310, 55313586130829865212025793302979452922870356482030868613037427298852922

Offset: 0

S. R. Kannan, Rajesh Kumar Mohapatra, Feb 29 2020

nonn

Also, the number of n X n distinct rooted fuzzy matrices.
The number of chains in the power set of n^2-elements such that the first term of the chains is either an empty set or a set of n^2-elements.
The number of chains in the collection of all binary (crisp or Boolean or logical) matrices of order n such that the first term of the chains is either null matrix or unit matrix.

S. R. Kannan and Rajesh Kumar Mohapatra, Counting the Number of Non-Equivalent Classes of Fuzzy Matrices Using Combinatorial Techniques, arXiv preprint arXiv:1909.13678 [math.GM], 2019.
V. Murali and B. Makamba, Finite Fuzzy Sets, Int. J. Gen. Syst., Vol. 34 (1) (2005), pp. 61-75.
R. B. Nelsen and H. Schmidt, Jr., Chains in power sets, Math. Mag., 64 (1) (1991), 23-31.
M. Tărnăuceanu, The number of chains of subgroups of a finite elementary abelian p-group, arXiv preprint arXiv:1506.08298 [math.GR], 2015.

Cf. A000629, A038719, A007047, A328044, A330301, A330302, A330804, A331957.

a(n) = A000629(n^2).

cp's OEIS Frontend

A260464 Number of chains in the poset of even-sized subsets of {1,2,...,n} ordered by inclusion.

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Examples

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Mathematica

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A260504 Number of chains in the poset of all odd-sized subsets of {1,2,...,n} ordered by inclusion.

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Maple

Mathematica

Formula

A292915 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)/(2 - exp(x)).

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A107954 Number of chains in the power set lattice, or the number of fuzzy subsets of an (n+4)-element set X_(n+4) with specification n elements of one kind, 3 elements of another and 1 of yet another kind.

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A330032 The number of chains of strictly rooted upper triangular or lower triangular matrices of order n.

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A365863 a(0) = 1; thereafter a(n) = n*Sum_{k = 0..n-1} binomial(n, k)*(-1)^(1+n+k)*a(k).

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A368317 Expansion of e.g.f. exp(4*x) / (2 - exp(x)).

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PARI

Formula

A107955 Number of chains in the power set lattice or the number of fuzzy subsets of an (n+5)-element set X_(n+5) with specification n elements of one kind, 4 elements of another and 1 of yet another kind.

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Author

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A365863 a(0) = 1; thereafter a(n) = nSum_{k = 0..n-1} binomial(n, k)(-1)^(1+n+k)*a(k).