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User: Daniel McLaury

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Daniel McLaury has authored 4 sequences.

A281097 Number of group homomorphisms S_3 -> S_n, where S_n denotes the symmetric group on n letters.

Original entry on oeis.org

1, 2, 10, 34, 146, 1036, 5692, 36380, 323164, 2394136, 19863416, 210278872, 1937685400

Offset: 1

Daniel McLaury, Apr 12 2017

A000085 gives the number of group homomorphisms S_2 -> S_n.

Cf. A378279, A378163.

List([1..8], n -> Length(AllHomomorphisms(SymmetricGroup(3), SymmetricGroup(n))));

a(n) = 6*A378279(n) + A000085(n). See A378163 for more information. - Jianing Song, Nov 27 2024

a(8) from Georg Fischer, Jun 16 2022

a(9)-a(13) from Jianing Song, Nov 27 2024

A165257 Triangle in which n-th row is binomial(n+k-1,k), for column k=1..n.

Original entry on oeis.org

1, 2, 3, 3, 6, 10, 4, 10, 20, 35, 5, 15, 35, 70, 126, 6, 21, 56, 126, 252, 462, 7, 28, 84, 210, 462, 924, 1716, 8, 36, 120, 330, 792, 1716, 3432, 6435, 9, 45, 165, 495, 1287, 3003, 6435, 12870, 24310

Offset: 1

Daniel McLaury (daniel.mclaury(AT)gmail.com), Sep 11 2009

T(n,k) is the number of non-descending sequences with length k and last number is less or equal to n. T(n,k) is also the number of integer partitions (of any positive integer) with length k and largest part is less or equal to n. - Zlatko Damijanic, Dec 06 2024

			1;
2, 3;
3, 6, 10;
4, 10, 20, 35;
5, 15, 35, 70, 126;
6, 21, 56, 126, 252, 462;
7, 28, 84, 210, 462, 924, 1716;

Harvey P. Dale, Table of n, a(n) for n = 1..1000

A059481 with the first column (k = 0) removed.

Cf. A030662 (row sums), A001700 (diagonal).

Table[Binomial[n+k-1,k],{n,10},{k,n}]//Flatten (* Harvey P. Dale, Jul 31 2021 *)

tabl(nn) = {for (n=1, nn, for (k=1, n, print1( binomial(n+k-1, k), ", ");); print(););} \\ Michel Marcus, Jun 12 2013

A144630 Triangle read by rows: T(n,k) (1 <= k <= n) is the sum of the entries in the lower right k X k submatrix of the n X n inverse Hilbert matrix.

Original entry on oeis.org

1, 12, 4, 180, 12, 9, 2800, 880, 40, 16, 44100, 46900, 4480, 40, 25, 698544, 1615824, 411264, 13104, 84, 36, 11099088, 45094896, 23653476, 2268756, 36036, 84, 49, 176679360, 1115345088, 1017615456, 207193536, 9660816, 79776, 144, 64

Offset: 1

Daniel McLaury and Ben Golub, Dec 23 2008

The initial entries in each row form A000515. The second entries give A144631. The final entries are the squares (A000290).

Row sums are A144632. The penultimate entries in each row appear to be 4*A014105. - N. J. A. Sloane, Jan 20 2009

			The first three inverse Hilbert matrices are:
--------------
[ 1 ]
--------------
[4 -6 ]
[-6 12]
--------------
[ 9 -36 30 ]
[-36 192 -180]
[30 -180 180]
--------------
Triangle begins:
1,
12, 4,
180, 12, 9,
2800, 880, 40, 16,
44100, 46900, 4480, 40, 25,
698544, 1615824, 411264, 13104, 84, 36

Klaus Brockhaus, Table of n, a(n) for n=1..1830 (rows 1 - 60)
Wikipedia, Hilbert matrix (gives inverse Hilbert matric explicitly).

Cf. A000290, A000515, A144631.

invhilb(1), invhilb(2), invhilb(3), etc.

&cat[ [ &+[I[i][j]: i,j in [k..n] ]: k in [n..1 by -1] ] where I:=H^-1 where H:=Matrix(Rationals(), n, n, [ < i, j, 1/(i+j-1) >: i, j in [1..n] ] ): n in [1..8] ]; // Klaus Brockhaus, Jan 21 2009

invH := proc(n,i,j) (-1)^(i+j)*(i+j-1)*binomial(n+i-1,n-j)*binomial(n+j-1,n-i)* (binomial(i+j-2,i-1))^2 ; end: A144630 := proc(n,k) local T,i,j ; T := 0 ; for i from n-k+1 to n do for j from n-k+1 to n do T := T+invH(n,i,j) ; od; od; RETURN(T) ; end: for n from 1 to 10 do for k from 1 to n do printf("%a,", A144630(n,k)) : od: od: # R. J. Mathar, Jan 21 2009

inverseHilbert[n_, i_, j_] := (-1)^(i+j)*(i+j-1) * Binomial[n+i-1, n-j] * Binomial[n+j-1, n-i] * Binomial[i+j-2, i-1]^2; inverseHilbert[n_, k_] := Table[ inverseHilbert[n, i, j], {i, n-k+1, n}, {j, n-k+1, n}]; t[n_, k_] := Tr[ Flatten[ inverseHilbert[n, k]]]; Flatten[ Table[t[n, k], {n, 1, 8}, {k, 1, n}]] (* Jean-François Alcover, Jul 16 2012 *)

More terms from R. J. Mathar and Klaus Brockhaus, Jan 21 2009

A144631 Second diagonal (or column) of A144630.

Original entry on oeis.org

4, 12, 880, 46900, 1615824, 45094896, 1115345088, 25519125060, 553014576400, 11514200107696, 232490008680384, 4581732884262352, 88532684825838400, 1683073282734360000, 31561148509363526400, 584964180982546208100

Offset: 2

Daniel McLaury and Ben Golub, Jan 20 2009

nonn

Klaus Brockhaus, Table of n, a(n) for n=2..100

[ [ &+[I[i][j]: i, j in [k..n] ]: k in [n..1 by -1] ][2] where I:=H^-1 where H:=Matrix(Rationals(), n, n, [ < i, j, 1/(i+j-1) >: i, j in [1..n] ] ): n in [2..17] ]; // Klaus Brockhaus, Jan 22 2009

invH := proc(n,i,j) (-1)^(i+j)*(i+j-1)*binomial(n+i-1,n-j)*binomial(n+j-1,n-i)* (binomial(i+j-2,i-1))^2 ; end: A144630 := proc(n,k) local T,i,j ; T := 0 ; for i from n-k+1 to n do for j from n-k+1 to n do T := T+invH(n,i,j) ; od; od; RETURN(T) ; end: A144631 := proc(n) A144630(n+1,2) ; end: for n from 1 to 20 do printf("%a,",A144631(n)) : od: # R. J. Mathar, Jan 21 2009

More terms from R. J. Mathar and Klaus Brockhaus, Jan 21 2009

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User: Daniel McLaury

A281097 Number of group homomorphisms S_3 -> S_n, where S_n denotes the symmetric group on n letters.

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A165257 Triangle in which n-th row is binomial(n+k-1,k), for column k=1..n.

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A144630 Triangle read by rows: T(n,k) (1 <= k <= n) is the sum of the entries in the lower right k X k submatrix of the n X n inverse Hilbert matrix.

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A144631 Second diagonal (or column) of A144630.

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