Steven Finch - cp's OEIS frontend

A376870 Reduced numerators of Newton's iteration for 1/sqrt(3), starting with 1/3.

1, 4, 130, 2739685, 21055737501685791580, 9337539302589041654242365815942422114384262970589593842110

Offset: 0

Steven Finch, Oct 07 2024

An explicit formula for a(n) is not known, although it arises from a recurrence and the corresponding denominators are simply 3^((3^n + 1)/2) = 3*A134799(n).

Next term is too large to include.

			a(1) = 4 because b(1) = (3/2)*(1/3)*(1 - 1/9) = 4/9.
1/3, 4/9, 130/243, 2739685/4782969, ... = A376870(n)/(3*A134799(n)).

Paolo Xausa, Table of n, a(n) for n = 0..7
X. Gourdon and P. Sebah, Pythagoras' Constant.

Cf. A020760, A134799, A376867.

Module[{n = 0}, NestList[#*(3^(3^n++ + 1) - #^2)/2 &, 1, 6]] (* Paolo Xausa, Oct 17 2024 *)

from itertools import count, islice
def A376870_gen(): # generator of terms
    p = 1
    for k in count(0):
        yield p
        p = p*(3**(3**k+1)-p**2)>>1
A376870_list = list(islice(A376870_gen(),6)) # Chai Wah Wu, Oct 11 2024

a(n) is the reduced numerator of b(n) = (3/2)*b(n-1)*(1 - b(n-1)^2); b(0) = 1/3.
Limit_{n -> oo} a(n)/(3*A134799(n)) = 1/sqrt(3) = A020760.
a(n+1) = a(n)*(3^(3^n+1)-a(n)^2)/2. - Chai Wah Wu, Oct 11 2024

A376867 Reduced numerators of Newton's iteration for 1/sqrt(2), starting with 1/2.

Original entry on oeis.org

1, 5, 355, 94852805, 1709678476417571835487555, 9994796326591347130392203807311551183419838794447313956622219314498503205

Offset: 0

Steven Finch, Oct 07 2024

An explicit formula for a(n) is not known, although it arises from a recurrence and the corresponding denominators are simply 2^(3^n) = A023365(n+1).

Next term is too large to include.

			a(1) = 5 because b(1) = (1/2)*(3/2 - 1/4) = 5/8.
1/2, 5/8, 355/512, 94852805/134217728, ... = a(n)/A023365(n+1).

Alois P. Heinz, Table of n, a(n) for n = 0..7
X. Gourdon and P. Sebah, Pythagoras' Constant.

Cf. A010503, A023365, A376870.

b:= proc(n) b(n):= `if`(n=0, 1/2, b(n-1)*(3/2-b(n-1)^2)) end:
a:= n-> numer(b(n)):
seq(a(n), n=0..5);  # Alois P. Heinz, Oct 07 2024

a[0]=1/2; a[n_]:=a[n-1](3/2-a[n-1]^2); Numerator[Array[a,6,0]] (* Stefano Spezia, Oct 15 2024 *)

from itertools import count, islice
def A376867_gen(): # generator of terms
    p = 1
    for k in count(0):
        yield p
        p *= ((3<<((3**k<<1)-1))-p**2)
A376867_list = list(islice(A376867_gen(),6)) # Chai Wah Wu, Oct 11 2024

a(n) is the reduced numerator of b(n) = b(n-1)*(3/2 - b(n-1)^2); b(0) = 1/2.
Limit_{n -> oo} a(n)/A023365(n+1) = 1/sqrt(2) = A010503.
a(n+1) = a(n)*(3*2^(2*3^n-1)-a(n)^2). - Chai Wah Wu, Oct 11 2024

A350276 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose fourth-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-3).

Original entry on oeis.org

1, 1, 4, 27, 255, 1, 3094, 1, 30, 45865, 46, 405, 340, 803424, 659, 3780, 10710, 4970, 16239720, 12867, 48405, 209440, 178920, 87864, 372076163, 284785, 1225665, 3005940, 5457060, 3558492, 1812384, 9529560676, 7126384, 32262300, 51205700, 135084600, 120593340, 81557280, 42609720

Offset: 0

Steven Finch, Dec 22 2021

An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.

If the mapping has no fourth component, then its fourth-smallest component is defined to have size 0.

			Triangle begins:
       1;
       1;
       4;
      27;
     255,   1;
    3094,   1,   30;
   45865,  46,  405,   340;
  803424, 659, 3780, 10710, 4970;
  ...

Alois P. Heinz, Rows n = 0..100, flattened
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Row sums give A000312.

Cf. A001865, A350078, A350079, A350080, A350081, A350275.

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, l) option remember; `if`(n=0, x^subs(infinity=0, l)[4],
      add(b(n-i, sort([l[], i])[1..4])*g(i)*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [infinity$4])):
seq(T(n), n=0..12);  # Alois P. Heinz, Dec 22 2021

g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, l_] := b[n, l] = If[n == 0, x^(l /. Infinity -> 0)[[4]], Sum[b[n - i, Sort[Append[l, i]][[1 ;; 4]]]*g[i]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := With[{p = b[n, Table[Infinity, {4}]]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

A350275 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose fourth-largest component has size exactly k; n >= 0, 0 <= k <= floor(n/4).

Original entry on oeis.org

1, 1, 4, 27, 255, 1, 3094, 31, 45865, 791, 803424, 20119, 16239720, 528991, 8505, 372076163, 14689441, 654885, 9529560676, 435580164, 34859160, 269819334245, 13846282341, 1646054025, 8369112382488, 471890017358, 73811825010, 1286223400

Offset: 0

Steven Finch, Dec 22 2021

An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.

If the mapping has no fourth component, then its fourth-largest component is defined to have size 0.

			Triangle begins:
       1;
       1;
       4;
      27;
     255,     1;
    3094,    31;
   45865,   791;
  803424, 20119;
  ...

Alois P. Heinz, Rows n = 0..100, flattened
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Row sums give A000312.

Cf. A001865, A350078, A350079, A350080, A350081, A350276.

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, l) option remember; `if`(n=0, x^l[1], add(g(i)*
      b(n-i, sort([l[], i])[-4..-1])*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$4])):
seq(T(n), n=0..14);  # Alois P. Heinz, Dec 22 2021

g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, l_] := b[n, l] = If[n == 0, x^l[[1]], Sum[g[i]*b[n - i, Sort[ Append[l, i]][[-4 ;; -1]]]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := With[{p = b[n, {0, 0, 0, 0}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

A350274 Triangle read by rows: T(n,k) is the number of n-permutations whose fourth-shortest cycle has length exactly k; n >= 0, 0 <= k <= max(0,n-3).

Original entry on oeis.org

1, 1, 2, 6, 23, 1, 109, 1, 10, 619, 16, 45, 40, 4108, 92, 210, 420, 210, 31240, 771, 1645, 2800, 2520, 1344, 268028, 6883, 17325, 15960, 26460, 18144, 10080, 2562156, 68914, 173250, 148400, 226800, 211680, 151200, 86400, 27011016, 757934, 1854930, 1798720, 1801800, 2494800, 1940400, 1425600, 831600

Offset: 0

Steven Finch, Dec 22 2021

If the permutation has no fourth cycle, then its fourth-longest cycle is defined to have length 0.

			Triangle begins:
[0]      1;
[1]      1;
[2]      2;
[3]      6;
[4]     23,    1;
[5]    109,    1,    10;
[6]    619,   16,    45,    40;
[7]   4108,   92,   210,   420,   210;
[8]  31240,  771,  1645,  2800,  2520,  1344;
[9] 268028, 6883, 17325, 15960, 26460, 18144, 10080;
    ...

Alois P. Heinz, Rows n = 0..100, flattened
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Column 0 is 1 for n=0, together with A000142(n) - A122105(n-1) for n>=1.
Row sums give A000142.
Cf. A126074, A145877, A332908, A349979, A349980, A350015, A350016, A350273.

m:= infinity:
b:= proc(n, l) option remember; `if`(n=0, x^`if`(l[4]=m,
      0, l[4]), add(b(n-j, sort([l[], j])[1..4])
               *binomial(n-1, j-1)*(j-1)!, j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [m$4])):
seq(T(n), n=0..11);  # Alois P. Heinz, Dec 22 2021

m = Infinity;
b[n_, l_] := b[n, l] = If[n == 0, x^If[l[[4]] == m, 0, l[[4]]], Sum[b[n-j, Sort[Append[l, j]][[1 ;; 4]]]*Binomial[n-1, j-1]*(j-1)!, {j, 1, n}]];
T[n_] := With[{p = b[n, {m, m, m, m}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 11}] // Flatten (* Jean-François Alcover, Dec 29 2021, after Alois P. Heinz *)

Sum_{k=0..n-3} k * T(n,k) = A332908(n) for n >= 4.

More terms from Alois P. Heinz, Dec 22 2021

A350273 Irregular triangle read by rows: T(n,k) is the number of n-permutations whose fourth-longest cycle has length exactly k; n >= 0, 0 <= k <= floor(n/4).

Original entry on oeis.org

1, 1, 2, 6, 23, 1, 109, 11, 619, 101, 4108, 932, 31240, 8975, 105, 268028, 91387, 3465, 2562156, 991674, 74970, 27011016, 11514394, 1391390, 311378616, 143188574, 24188010, 246400, 3897004032, 1905067958, 412136010, 12812800, 52626496896, 27059601596, 7053834788, 438357920

Offset: 0

Steven Finch, Dec 22 2021

If the permutation has no fourth cycle, then its fourth-longest cycle is defined to have length 0.

			Triangle begins:
[0]      1;
[1]      1;
[2]      2;
[3]      6;
[4]     23,     1;
[5]    109,    11;
[6]    619,   101;
[7]   4108,   932;
[8]  31240,  8975,  105;
[9] 268028, 91387, 3465;
    ...

Alois P. Heinz, Rows n = 0..100, flattened
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Row sums give A000142(n).

Cf. A126074, A145877, A332853, A349979, A349980, A350015, A350016, A350274.

b:= proc(n, l) option remember; `if`(n=0, x^l[1], add((j-1)!*
      b(n-j, sort([l[], j])[2..5])*binomial(n-1, j-1), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$4])):
seq(T(n), n=0..14);  # Alois P. Heinz, Dec 22 2021

b[n_, l_] := b[n, l] = If[n == 0, x^l[[1]], Sum[(j - 1)!*b[n - j, Sort[ Append[l, j]][[2 ;; 5]]]*Binomial[n - 1, j - 1], {j, 1, n}]];
T[n_] := With[{p = b[n, {0, 0, 0, 0}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 29 2021, after Alois P. Heinz *)

Sum_{k=0..floor(n/4)} k * T(n,k) = A332853(n) for n >= 4.

More terms from Alois P. Heinz, Dec 22 2021

A350081 Triangle read by rows: T(n,k) is the number of endofunctions on [n] whose third-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-2).

Original entry on oeis.org

1, 1, 4, 26, 1, 237, 1, 18, 2789, 31, 135, 170, 40270, 386, 810, 3060, 2130, 689450, 6574, 13545, 36295, 44730, 32949, 13657756, 129291, 327285, 323680, 944300, 790776, 604128, 307348641, 2910709, 7207137, 6602120, 15476580, 18780930, 16311456, 12782916

Offset: 0

Steven Finch, Dec 12 2021

An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.

If the mapping has no third component, then its third-smallest component is defined to have size 0.

			Triangle begins:
       1;
       1;
       4;
      26,    1;
     237,    1,    18;
    2789,   31,   135,   170;
   40270,  386,   810,  3060,  2130;
  689450, 6574, 13545, 36295, 44730, 32949;
  ...

Alois P. Heinz, Rows n = 0..120, flattened
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Row sums give A000312.

Cf. A001865, A350078, A350079, A350080, A350275, A350276.

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, l) option remember; `if`(n=0, x^subs(infinity=0, l)[3],
      add(b(n-i, sort([l[], i])[1..3])*g(i)*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [infinity$3])):
seq(T(n), n=0..12);  # Alois P. Heinz, Dec 17 2021

g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, l_] := b[n, l] = If[n == 0, x^(l /. Infinity -> 0)[[3]], Sum[b[n - i, Sort[Append[l, i]][[1 ;; 3]]]*g[i]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := With[{p = b[n, {Infinity, Infinity, Infinity}]}, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

More terms (two rows) from Alois P. Heinz, Dec 16 2021

A350080 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose third-largest component has size exactly k; n >= 0, 0 <= k <= floor(n/3).

Original entry on oeis.org

1, 1, 4, 26, 1, 237, 19, 2789, 336, 40270, 5981, 405, 689450, 115193, 18900, 13657756, 2459955, 659505, 307348641, 58366045, 20330163, 1375640, 7745565616, 1530739594, 623758590, 99936200, 216114310994, 44076571672, 19795671225, 5325116720

Offset: 0

Steven Finch, Dec 12 2021

An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.

If the mapping has no third component, then its third-largest component is defined to have size 0.

			Triangle begins:
       1;
       1;
       4;
      26,     1;
     237,    19;
    2789,   336;
   40270,   5981,   405;
  689450, 115193, 18900;
  ...

Alois P. Heinz, Rows n = 0..120, flattened
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Row sums give A000312.

Cf. A001865, A350078, A350079, A350081, A350275, A350276.

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, l) option remember; `if`(n=0, x^l[1], add(g(i)*
      b(n-i, sort([l[], i])[-3..-1])*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$3])):
seq(T(n), n=0..12);  # Alois P. Heinz, Dec 17 2021

g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, l_] := b[n, l] = If[n == 0, x^l[[1]], Sum[g[i]*b[n - i, Sort[ Append[l, i]][[-3 ;; -1]]]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := With[{p = b[n, {0, 0, 0}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

More terms (4 rows) from Alois P. Heinz, Dec 16 2021

A350079 Triangle read by rows: T(n,k) is the number of endofunctions on [n] whose second-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-1).

Original entry on oeis.org

1, 1, 3, 1, 17, 1, 9, 142, 19, 27, 68, 1569, 201, 135, 510, 710, 21576, 2921, 3465, 2890, 6390, 9414, 355081, 50233, 63630, 20230, 84490, 98847, 151032, 6805296, 1004599, 1196181, 918680, 705740, 1493688, 1812384, 2840648, 148869153, 22872097, 26904339, 23943752, 6351660, 28072548, 30810528, 38348748, 61247664

Offset: 0

Steven Finch, Dec 12 2021

An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.

If the mapping has no second component, then its second-smallest component is defined to have size 0.

			Triangle begins:
       1;
       1;
       3,     1;
      17,     1,     9;
     142,    19,    27,    68;
    1569,   201,   135,   510,   710;
   21576,  2921,  3465,  2890,  6390,  9414;
  355081, 50233, 63630, 20230, 84490, 98847, 151032;
  ...

Alois P. Heinz, Rows n = 0..141, flattened
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Column 0 gives gives 1 together with A001865.
Row sums give A000312.
Cf. A001865, A350078, A350080, A350081, A350275, A350276

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, l) option remember; `if`(n=0, x^subs(infinity=0, l)[2],
      add(b(n-i, sort([l[], i])[1..2])*g(i)*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [infinity$2])):
seq(T(n), n=0..12);  # Alois P. Heinz, Dec 17 2021

g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, l_] := b[n, l] = If[n == 0, x^(l /. Infinity -> 0)[[2]], Sum[b[n - i, Sort[Append[l, i]][[1;;2]]]*g[i]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := With[{p = b[n, {Infinity, Infinity}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

More terms (two rows) from Alois P. Heinz, Dec 15 2021

A350078 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose second-largest component has size exactly k; n >= 0, 0 <= k <= floor(n/2).

Original entry on oeis.org

1, 1, 3, 1, 17, 10, 142, 87, 27, 1569, 911, 645, 21576, 11930, 10260, 2890, 355081, 189610, 174132, 104720, 6805296, 3543617, 3229275, 2493288, 705740, 148869153, 76060087, 67843521, 60223520, 34424208, 3660215680, 1842497914, 1605373560, 1530575960, 1051155000, 310181886

Offset: 0

Steven Finch, Dec 12 2021

An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.

If the mapping has no second component, then its second-largest component is defined to have size 0.

			Triangle begins:
       1;
       1;
       3,      1;
      17,     10;
     142,     87,     27;
    1569,    911,    645;
   21576,  11930,  10260,   2890;
  355081, 189610, 174132, 104720;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Column 0 gives gives 1 together with A001865.
Row sums give A000312.
Cf. A001865, A350079, A350080, A350081, A350275, A350276.

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, l) option remember; `if`(n=0, x^l[1], add(g(i)*
      b(n-i, sort([l[], i])[-2..-1])*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$2])):
seq(T(n), n=0..10);  # Alois P. Heinz, Dec 17 2021

g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, l_] := g[n, l] = If[n == 0, x^l[[1]], Sum[g[i]*b[n - i, Sort[ Append[l, i]][[-2 ;; -1]]]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := With[{p = b[n, {0, 0}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

More terms (three rows) from Alois P. Heinz, Dec 15 2021

cp's OEIS Frontend

User: Steven Finch

A376870 Reduced numerators of Newton's iteration for 1/sqrt(3), starting with 1/3.

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A376867 Reduced numerators of Newton's iteration for 1/sqrt(2), starting with 1/2.

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A350276 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose fourth-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-3).

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A350275 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose fourth-largest component has size exactly k; n >= 0, 0 <= k <= floor(n/4).

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A350274 Triangle read by rows: T(n,k) is the number of n-permutations whose fourth-shortest cycle has length exactly k; n >= 0, 0 <= k <= max(0,n-3).

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A350273 Irregular triangle read by rows: T(n,k) is the number of n-permutations whose fourth-longest cycle has length exactly k; n >= 0, 0 <= k <= floor(n/4).

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A350081 Triangle read by rows: T(n,k) is the number of endofunctions on [n] whose third-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-2).

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