A049353 -id:A049353 - cp's OEIS frontend

A049378 Row sums of triangle A049353.

1, 1, 6, 46, 436, 4956, 65776, 996976, 16957536, 319259296, 6581662336, 147290942976, 3552885191296, 91827536814976, 2530228890080256, 74003737259670016, 2288810287491774976, 74607500831801289216, 2555587654482227055616, 91746983502042106018816

Offset: 0

Wolfdieter Lang

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..421
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Column k=5 of A291709.

Cf. A049353, A049426.

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*(j+3)!/4!*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 01 2017

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, j - 1]*(j + 3)!/4!*a[n - j], {j, 1, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

E.g.f. exp(p(x)) with p(x) := x*(2-x)*(2-2*x+x^2)/(4*(1-x)^4) (E.g.f. first column of A049353).
From Seiichi Manyama, Jan 18 2025: (Start)
a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A004213(k).
a(n) = (1/exp(1/4)) * (-1)^n * n! * Sum_{k>=0} binomial(-4*k,n)/(4^k * k!). (End)

a(0)=1 prepended by Alois P. Heinz, Aug 01 2017

A144355 Partition number array, called M31(5), related to A049353(n,m)= |S1(5;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 5, 1, 30, 15, 1, 210, 120, 75, 30, 1, 1680, 1050, 1500, 300, 375, 50, 1, 15120, 10080, 15750, 9000, 3150, 9000, 1875, 600, 1125, 75, 1, 151200, 105840, 176400, 220500, 35280, 110250, 63000, 78750, 7350, 31500, 13125, 1050, 2625, 105, 1, 1663200, 1209600, 2116800

Offset: 1

Wolfdieter Lang Oct 09 2008, Oct 28 2008

Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M31(5;n,k) with the k-th partition of n in A-St order.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
Fifth member (K=5) in the family M31(K) of partition number arrays.
If M31(5;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle |S1(5)|:= A049353.

			[1];[5,1];[30,15,1];[210,120,75,30,1];[1680,1050,1500,300,375,50,1];...
a(4,3)= 75 = 3*|S1(5;2,1)|^2. The relevant partition of 4 is (2^2).

W. Lang, First 10 rows of the array and more.
W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

A049378 (row sums).

A144354 (M31(4) array), A144356 (M31(6) array).

a(n,k)=(n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S1(5;j,1)|^e(n,k,j),j=1..n)= M3(n,k)*product(|S1(5;j,1)|^e(n,k,j),j=1..n) with |S1(5;n,1)|= A001720(n+3) = (n+3)!/4!, n>=1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. M3(n,k)=A036040.

A134139 Alternating row sums of triangle A049353 (S1p(5)).

Original entry on oeis.org

1, 4, 16, 44, -244, -7336, -112664, -1406336, -14058944, -66627136, 1879992896, 85070733824, 2359547577856, 54106723270144, 1055100890838016, 15611883969810944, 48348082318102016, -9055743653514520576, -521351156412528902144

Offset: 1

Wolfdieter Lang Oct 12 2007

Cf. A049378 (row sums of A049353).

a(n)=sum(A049353(n,m)*(-1)^(m-1),m=1..n), n>=1.

E.g.f.: 1-exp(-x*(2-x)*(2-2*x+x^2)/(4*(1-x)^4)). Cf. e.g.f. first column of A049353.

A001720 a(n) = n!/24.

Original entry on oeis.org

1, 5, 30, 210, 1680, 15120, 151200, 1663200, 19958400, 259459200, 3632428800, 54486432000, 871782912000, 14820309504000, 266765571072000, 5068545850368000, 101370917007360000, 2128789257154560000, 46833363657400320000, 1077167364120207360000

Offset: 4

N. J. A. Sloane

The asymptotic expansion of the higher-order exponential integral E(x,m=1,n=5) ~ exp(-x)/x*(1 - 5/x + 30/x^2 - 210/x^3 + 1680/x^4 - 15120/x^5 + 151200/x^6 - 1663200/x^7 + ...) leads to this sequence. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 4..300
S. Cerdá, A simple scheme to find the solutions to Brocard-Ramanujan Diophantine Equation n! + 1 = m^2, arXiv:1504.06694 [math.NT], 2015.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 264.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
Index entries for sequences related to factorial numbers.
Index to divisibility sequences.

Cf. A000142, A049353, A049459, A051338, A245334.
Cf. A001710, A001715, A007696, A094638.
Cf. A094587, A130534, A138533, A163931, A173333, A213936.

a001720 = (flip div 24) . a000142 -- Reinhard Zumkeller, Aug 31 2014

[Factorial(n)/24: n in [4..25]]; // Vincenzo Librandi, Jul 20 2011

a[n_]:=n!/24; (* Vladimir Joseph Stephan Orlovsky, Dec 13 2008 *)
Range[4,30]!/24 (* Harvey P. Dale, Jul 27 2021 *)

a(n)=n!/24 \\ Charles R Greathouse IV, Jan 12 2012

a(n)= A049353(n-3, 1) (first column of triangle).
E.g.f. if offset 0: 1/(1-x)^5.
a(n) = A173333(n,4). - Reinhard Zumkeller, Feb 19 2010
a(n) = A245334(n,n-4) / 5. - Reinhard Zumkeller, Aug 31 2014
G(x) = (1 - (1 + x)^(-4)) / 4 = x - 5 x^2/2! + 30 x^3/3! - ..., an e.g.f. for this signed sequence (for n!/4!), is the compositional inverse of H(x) = (1 - 4*x)^(-1/4) - 1 = x + 5 x^2/2! + 45 x^3/3! + ..., an e.g.f. for A007696. Cf. A094638, A001710 (for n!/2!), and A001715 (for n!/3!). Cf. columns of A094587, A173333, and A213936 and rows of A138533. - Tom Copeland, Dec 27 2019
E.g.f.: x^4 / (4! * (1 - x)). - Ilya Gutkovskiy, Jul 09 2021
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=4} 1/a(n) = 24*e - 64.
Sum_{n>=4} (-1)^n/a(n) = 24/e - 8. (End)

A049374 A triangle of numbers related to triangle A030527.

Original entry on oeis.org

1, 6, 1, 42, 18, 1, 336, 276, 36, 1, 3024, 4200, 960, 60, 1, 30240, 66024, 23400, 2460, 90, 1, 332640, 1086624, 557424, 87360, 5250, 126, 1, 3991680, 18805248, 13349952, 2916144, 255360, 9912, 168, 1, 51891840, 342486144, 325854144, 95001984

Offset: 1

Wolfdieter Lang

a(n,1) = A001725(n+4). a(n,m)=: S1p(6; n,m), a member of a sequence of lower triangular Jabotinsky matrices with nonnegative entries, including S1p(1; n,m) = A008275 (unsigned Stirling first kind), S1p(2; n,m) = A008297(n,m) (unsigned Lah numbers). S1p(3; n,m) = A046089(n,m), S1p(4; n,m) = A049352, S1p(5; n,m) = A049353(n,m).
Signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A049385(n,m) =: S2(6; n,m). The monic row polynomials E(n,x) := Sum_{m=1..n} (a(n,m)*x^m), E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
a(n,m) enumerates unordered increasing n-vertex m-forests composed of m unary trees (out-degree r from {0,1}) whose vertices of depth (distance from the root) j >= 1 come in j+5 colors. The k roots (j=0) each come in one (or no) color. - Wolfdieter Lang, Oct 12 2007

			Triangle begins
       1;
       6,       1;
      42,      18,      1;
     336,     276,     36,     1;
    3024,    4200,    960,    60,    1;
   30240,   66024,  23400,  2460,   90,   1;
  332640, 1086624, 557424, 87360, 5250, 126, 1;
E.g., row polynomial E(3,x) = 42*x + 18*x^2 + x^3.
a(4,2) = 276 = 4*(6*7) + 3*(6*6) from the two types of unordered 2-forests of unary increasing trees associated with the two m=2 parts partitions (1,3) and (2^2) of n=4. The first type has 4 increasing labelings, each coming in (1)*(1*6*7)=42 colored versions, e.g., ((1c1),(2c1,3c6,4c3)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 6 colors, c1..c6, can be chosen and the vertex labeled 4 with j=2 can come in 7 colors, e.g., c1..c7. Therefore there are 4*((1)*(1*6*7))=168 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*6)*(1*6))=108 such forests, e.g., ((1c1,3c4)(2c1,4c6)) or ((1c1,3c5)(2c1,4c2)), etc. - _Wolfdieter Lang_, Oct 12 2007

Muniru A Asiru, Table of n, a(n) for n = 1..1275
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
W. Lang, First ten rows.

Cf. A049402 (row sums), A134140 (alternating row sums).

Flat(List([1..10],n->Factorial(n)*List([1..n],k->Sum([1..k],j->(-1)^(k-j)*Binomial(k,j)*Binomial(n+5*j-1,5*j-1)/(5^k*Factorial(k)))))); # Muniru A Asiru, Jun 23 2018

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (n+5)!/120, 10); # Peter Luschny, Jan 28 2016

a[n_, k_] = n!*Sum[(-1)^(k-j)*Binomial[k, j]*Binomial[n + 5j - 1, 5j - 1]/(5^k*k!), {j, 1, k}] ;
Flatten[Table[a[n, k], {n, 1, 9}, {k, 1, n}] ][[1 ;; 40]]
(* Jean-François Alcover, Jun 01 2011, after Vladimir Kruchinin *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 10;
M = BellMatrix[(#+5)!/120&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

a(n,k)=(n!*sum((-1)^(k-j)*binomial(k,j)*binomial(n+5*j-1,5*j-1),j,1,k))/(5^k*k!); /* Vladimir Kruchinin, Apr 01 2011 */

a(n,k)=(n!*sum(j=1,k,(-1)^(k-j)*binomial(k,j)*binomial(n+5*j-1,5*j-1)))/(5^k*k!);
for(n=1,12,for(k=1,n,print1(a(n,k),", "));print()); /* print triangle */ /* Joerg Arndt, Apr 01 2011 */

a(n, m) = n!*A030527(n, m)/(m!*5^(n-m)); a(n, m) = (5*m+n-1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1; a(n, m)=0, n < m; a(n, 0) := 0; a(1, 1)=1. E.g.f. for m-th column: ((x*(5 - 10*x + 10*x^2 - 5*x^3 + x^4)/(5*(1-x)^5))^m)/m!.

a(n,k) = n!* Sum_{j=1..k} (-1)^(k-j)*binomial(k,j)*binomial(n+5*j-1,5*j-1) /(5^k*k!). - Vladimir Kruchinin, Apr 01 2011

A157385 A partition product of Stirling_1 type [parameter k = -5] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 5, 1, 15, 30, 1, 105, 120, 210, 1, 425, 1800, 1050, 1680, 1, 3075, 18600, 18900, 10080, 15120, 1, 15855, 174300, 338100, 211680, 105840, 151200, 1, 123515, 2227680, 4865700, 4327680, 2540160, 1209600, 1663200, 1, 757755

Offset: 1

Peter Luschny, Mar 07 2009, Mar 14 2009

Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = -5,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144355.
Same partition product with length statistic is A049353.
Diagonal a(A000217(n)) = rising_factorial(5,n-1), A001720(n+3).
Row sum is A049378.

Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_1 Triangles.

Cf. A157386, A157384, A157383, A157400, A126074, A157391, A157392, A157393, A157394, A157395

T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n
T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that
1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-2}(j-n-3).

A144891 Lower triangular array called S1hat(5) related to partition number array A144890.

Original entry on oeis.org

1, 5, 1, 30, 5, 1, 210, 55, 5, 1, 1680, 360, 55, 5, 1, 15120, 3630, 485, 55, 5, 1, 151200, 29820, 4380, 485, 55, 5, 1, 1663200, 321300, 39570, 5005, 485, 55, 5, 1, 19958400, 3225600, 421800, 43320, 5005, 485, 55, 5, 1, 259459200, 38808000, 4265100, 470550, 46445, 5005

Offset: 1

Wolfdieter Lang Oct 09 2008

If in the partition array M31hat(5):=A144890 entries with the same parts number m are summed one obtains this triangle of numbers S1hat(5). In the same way the signless Stirling1 triangle |A008275| is obtained from the partition array M_2 = A036039.

The first columns are A001720(n+3)=(n+3)!/4!, A144893, A144894,...

			[1];[5,1];[30,5,1];[210,55,5,1];[1680,360,55,5,1];...

W. Lang, First 10 rows of the array and more.
W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

A144892 (row sums).

a(n,m)=sum(product(|S1(5;j,1)|^e(n,m,q,j),j=1..n),q=1..p(n,m)) if n>=m>=1, else 0. Here p(n,m)=A008284(n,m), the number of m parts partitions of n and e(n,m,q,j) is the exponent of j in the q-th m part partition of n. |S1(5,n,1)|= A049353(n,1) = A001720(n+3) = (n+3)!/4!.

A134141 Generalized unsigned Stirling1 triangle, S1p(7).

Original entry on oeis.org

1, 7, 1, 56, 21, 1, 504, 371, 42, 1, 5040, 6440, 1295, 70, 1, 55440, 114520, 36225, 3325, 105, 1, 665280, 2116800, 983920, 135975, 7105, 147, 1, 8648640, 40884480, 26714800, 5199145, 398860, 13426, 196, 1, 121080960, 826338240, 735469280

Offset: 1

Wolfdieter Lang, Oct 12 2007

Signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A092082(n, m) =: S2(7; n,m). The monic row polynomials E(n,x) := sum(a(n,m)*x^m, m=1..n), E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
a(n,m) enumerates unordered increasing n-vertex m-forests composed of m unary trees (out-degree r from {0,1}) whose vertices of depth (distance from the root) j>=1 come in j+6 colors. The k roots (j=0) each come in one (or no) color. - Wolfdieter Lang, Oct 05 2007
A triangle of numbers related to triangle A132166.
a(n,1)= A001730(n,5), n>=1. a(n,m)=: S1p(7; n,m), a member of a sequence of lower triangular Jabotinsky matrices with nonnegative entries, including S1p(1; n, m)= A008275 (unsigned Stirling first kind), S1p(2; n,m)= A008297(n, m) (unsigned Lah numbers). S1p(3; n,m)= A046089(n,m), S1p(4; n,m)= A049352, S1p(5; n,m)= A049353(n,m), S1p(6; n,m)= A049374(n, m).
The Bell transform of factorial(n+6)/factorial(6). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

			{1}; {7,1}; {56,21,1}; {504,371,42,1}; ... E.g. Row polynomial E(3,x)=56*x+21*x^2+x^3.
a(4,2)= 371 = 4*(7*8)+3*(7*7) from the two types of unordered 2-forests of unary increasing trees associated with the two m=2 parts partitions (1,3) and (2^2) of n=4. The first type has 4 increasing labelings, each coming in (1)*(1*7*8)=56 colored versions, e.g., ((1c1),(2c1,3c7,4c5)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 7 colors, c1..c7, can be chosen and the vertex labeled 4 with j=2 can come in 8 colors, e.g., c1..c8. Therefore there are 4*((1)*(1*7*8))=224 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*7)*(1*7))=147 such forests, e.g. ((1c1,3c4)(2c1,4c7)) or ((1c1,3c6)(2c1,4c2)), etc. - _Wolfdieter Lang_, Oct 05 2007

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
W. Lang, First ten rows.

First column A001730(n+5), n>=1.

Row sums A132164. Alternating row sums A132165.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (n+6)!/6!, 9); # Peter Luschny, Jan 27 2016

a[n_, m_] /; n >= m >= 1 := a[n, m] = (6*m + n - 1)*a[n-1, m] + a[n-1, m-1]; a[n_, m_] /; n < m = 0; a[, 0] = 0; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]][[1 ;; 39]] (* _Jean-François Alcover, Jun 01 2011, after formula *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[(# + 6)!/6! &, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: factorial(n+6)/factorial(6), 10) # Peter Luschny, Jan 18 2016

a(n, m) = n!*A132166(n, m)/(m!*6^(n-m)); a(n, m) = (6*m+n-1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1; a(n, m)=0, n

A144890 Partition number array, called M31hat(5).

Original entry on oeis.org

1, 5, 1, 30, 5, 1, 210, 30, 25, 5, 1, 1680, 210, 150, 30, 25, 5, 1, 15120, 1680, 1050, 900, 210, 150, 125, 30, 25, 5, 1, 151200, 15120, 8400, 6300, 1680, 1050, 900, 750, 210, 150, 125, 30, 25, 5, 1, 1663200, 151200, 75600, 50400, 44100, 15120, 8400, 6300, 5250, 4500

Offset: 1

Wolfdieter Lang Oct 09 2008, Oct 28 2008

Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M31hat(5;n,k) with the k-th partition of n in A-St order.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
Fourth member (K=5) in the family M31hat(K) of partition number arrays.
If M31hat(5;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1hat(5):= A144891.

			[1];[5,1];[30,5,1];[210,30,25,5,1];[1680,210,150,30,25,5,1];...
a(4,3)= 25 = |S1(5;2,1)|^2. The relevant partition of 4 is (2^2).

W. Lang, First 10 rows of the array and more.
W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

Cf. A144892 (row sums).

Cf. A144885 (M31hat(4) array). A144891 (S1hat(5)).

a(n,k) = product(|S1(5;j,1)|^e(n,k,j),j=1..n) with |S1(5;n,1)| = A049353(n,1) = A001720(n+3) = [1,5,30,210,1680,...] = (n+3)!/4!, n>=1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n.

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A049378 Row sums of triangle A049353.

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A144355 Partition number array, called M31(5), related to A049353(n,m)= |S1(5;n,m)| (generalized Stirling triangle).

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A134139 Alternating row sums of triangle A049353 (S1p(5)).

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A001720 a(n) = n!/24.

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A049374 A triangle of numbers related to triangle A030527.

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A157385 A partition product of Stirling_1 type [parameter k = -5] with biggest-part statistic (triangle read by rows).

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A144891 Lower triangular array called S1hat(5) related to partition number array A144890.

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A134141 Generalized unsigned Stirling1 triangle, S1p(7).

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