A007799 -id:A007799 - cp's OEIS frontend

A004070 Table of Whitney numbers W(n,k) read by antidiagonals, where W(n,k) is maximal number of pieces into which n-space is sliced by k hyperplanes, n >= 0, k >= 0.

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 4, 1, 1, 2, 4, 7, 5, 1, 1, 2, 4, 8, 11, 6, 1, 1, 2, 4, 8, 15, 16, 7, 1, 1, 2, 4, 8, 16, 26, 22, 8, 1, 1, 2, 4, 8, 16, 31, 42, 29, 9, 1, 1, 2, 4, 8, 16, 32, 57, 64, 37, 10, 1, 1, 2, 4, 8, 16, 32, 63, 99, 93, 46, 11, 1, 1, 2, 4, 8, 16, 32, 64, 120, 163

Offset: 0

N. J. A. Sloane

As a number triangle, this is given by T(n,k)=sum{j=0..n, C(n,j)(-1)^(n-j)sum{i=0..j, C(j+k,i-k)}}. - Paul Barry, Aug 23 2004
As a number triangle, this is the Riordan array (1/(1-x), x(1+x)) with T(n,k)=sum{i=0..n, binomial(k,i-k)}. Diagonal sums are then A023434(n+1). - Paul Barry, Feb 16 2005
Form partial sums across rows of square array of binomial coefficients A026729; see also A008949. - Philippe Deléham, Aug 28 2005
Square array A026729 -> Partial sums across rows
1 0 0 0 0 0 0 . . . . 1 1 1 1 1 1 1 . . . . . .
1 1 0 0 0 0 0 . . . . 1 2 2 2 2 2 2 . . . . . .
1 2 1 0 0 0 0 . . . . 1 3 4 4 4 4 4 . . . . . .
1 3 3 1 0 0 0 . . . . 1 4 7 8 8 8 8 . . . . . .
For other Whitney numbers see A007799.
W(n,k) is the number of length k binary sequences containing no more than n 1's. - Geoffrey Critzer, Mar 15 2010
From Emeric Deutsch, Jun 15 2010: (Start)
Viewed as a number triangle, T(n,k) is the number of internal nodes of the Fibonacci tree of order n+2 at level k. A Fibonacci tree of order n (n>=2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node.
(End)
Named after the American mathematician Hassler Whitney (1907-1989). - Amiram Eldar, Jun 13 2021

			Table W(n,k) begins:
  1 1 1 1  1  1  1 ...
  1 2 3 4  5  6  7 ...
  1 2 4 7 11 16 22 ...
  1 2 4 8 15 26 42 ...
W(2,4) = 11 because there are 11 length 4 binary sequences containing no more than 2 1's: {0, 0, 0, 0}, {0, 0, 0, 1}, {0, 0, 1, 0}, {0, 0, 1, 1}, {0, 1, 0, 0}, {0, 1, 0, 1}, {0, 1, 1, 0}, {1, 0, 0, 0}, {1, 0, 0, 1}, {1, 0, 1, 0}, {1, 1, 0, 0}. - _Geoffrey Critzer_, Mar 15 2010
Table T(n, k) begins:
  1
  1  1
  1  2  1
  1  2  3  1
  1  2  4  4  1
  1  2  4  7  5  1
  1  2  4  8 11  6  1
...

Donald E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.

Gustav Burosch, Hans-Dietrich O.F. Gronau, Jean-Marie Laborde and Ingo Warnke, On posets of m-ary words, Discrete Math., Vol. 152, No. 1-3 (1996), pp. 69-91. MR1388633 (97e:06002)
Matteo Cervetti and Luca Ferrari, Pattern avoidance in the matching pattern poset, arXiv:2009.01024 [math.CO], 2020.
Matteo Cervetti and Luca Ferrari, Enumeration of Some Classes of Pattern Avoiding Matchings, with a Glimpse into the Matching Pattern Poset, Annals of Combinatorics (2022).
Richard K. Guy, Letter to N. J. A. Sloane, Apr 1975.
Yasuichi Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, Vol. 20, No. 2 (1982), pp. 168-178.
Robin Pemantle and Mark C. Wilson, Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions, SIAM Rev., Vol. 50, No. 2 (2008), pp. 199-272. See p. 233.

Cf. A007799. As a triangle, mirror A052509.
Rows converge to powers of two (A000079). Subdiagonals include A000225, A000295, A002662, A002663, A002664, A035038, A035039, A035040, A035041, A035042. Antidiagonal sums are A000071.
Rows are: A000012, A000027, A000124, A000125, A000127, A006261, A008859, A008860, A008861, A008862, A008863.
Cf. A178522, A178524.

Transpose[ Table[Table[Sum[Binomial[n, k], {k, 0, m}], {m, 0, 15}], {n, 0, 15}]] // Grid (* Geoffrey Critzer, Mar 15 2010 *)
T[ n_, k_] := Sum[ Binomial[n, j] (-1)^(n - j) Sum[ Binomial[j + k, i - k], {i, 0, j}], {j, 0, n}]; (* Michael Somos, May 31 2016 *)

/* array read by antidiagonals up coordinate index functions */
t1(n) = binomial(floor(3/2 + sqrt(2+2*n)), 2) - (n+1); /* A025581 */
t2(n) = n - binomial(floor(1/2 + sqrt(2+2*n)), 2); /* A002262 */
/* define the sequence array function for A004070 */
W(n, k) = sum(i=0, n, binomial(k, i));
/* visual check ( origin 0,0 ) */
printp(matrix(7, 7, n, k, W(n-1, k-1)));
/* print the sequence entries by antidiagonals going up ( origin 0,0 ) */
print1("S A004070 "); for(n=0, 32, print1(W(t1(n), t2(n))","));
print1("T A004070 "); for(n=33, 61, print1(W(t1(n), t2(n))","));
print1("U A004070 "); for(n=62, 86, print1(W(t1(n), t2(n))",")); /* Michael Somos, Apr 28 2000 */

T(n, k)=sum(m=0, n-k, binomial(k, m)) \\ Jianing Song, May 30 2022

W(n, k) = Sum_{i=0..n} binomial(k, i). - Bill Gosper
W(n, k) = if k=0 or n=0 then 1 else W(n, k-1)+W(n-1, k-1). - David Broadhurst, Jan 05 2000
The table W(n,k) = A000012 * A007318(transform), where A000012 = (1; 1,1; 1,1,1; ...). - Gary W. Adamson, Nov 15 2007
E.g.f. for row n: (1 + x + x^2/2! + ... + x^n/n!)* exp(x). - Geoffrey Critzer, Mar 15 2010
G.f.: 1 / (1 - x - x*y*(1 - x^2)) = Sum_{0 <= k <= n} x^n * y^k * T(n, k). - Michael Somos, May 31 2016
W(n, n) = 2^n. - Michael Somos, May 31 2016
From Jianing Song, May 30 2022: (Start)
T(n, 0) = T(n, n) = 1 for n >= 0; T(n, k) = T(n-1, k-1) + T(n-2, k-1) for k=1, 2, ..., n-1, n >= 2.
T(n, k) = Sum_{m=0..n-k} binomial(k, m).
T(n,k) = 2^k for 0 <= k <= floor(n/2). (End)

More terms from Larry Reeves (larryr(AT)acm.org), Mar 20 2000

A192837 Molecular topological indices of the permutation star graphs.

Original entry on oeis.org

0, 4, 132, 4680, 214080, 12416400, 896132160, 79295610240, 8481591336960, 1081908144172800, 162548813750400000, 28443681284170521600, 5739117489117031219200, 1323378125974080765388800, 345972881092262536240128000, 101817548412839690547916800000

Offset: 1

Eric W. Weisstein, Jul 11 2011

nonn

The permutation star graph of order n is a vertex transitive graph with n! vertices and degree n-1. The graph can be constructed as the Cayley graph of the permutations of 1..n with the n-1 generators (1 2), (1 3)..(1 n) where (1 k) is the transposition of 1 and k. The number of nodes at distance k from a specified node is given by A007799(n,k). - Andrew Howroyd, May 13 2017

Andrew Howroyd, Table of n, a(n) for n = 1..50
Eric Weisstein's World of Mathematics, Molecular Topological Index
Eric Weisstein's World of Mathematics, Permutation Star Graph

Cf. A007799.

a[n_, 0] = 1; a[n_, 1] = n - 1; a[n_, 2] = (n - 1) (n - 2);
a[n_, k_ /; k >= 2] := a[n, k] = (n - 1) a[n - 1, k - 1] + Sum[j a[j, k - 3], {j, n - 2}];
Table[n! (n - 1) (n - 1 + Sum[k a[n, k], {k, Floor[3 (n - 1)/2]}]), {n, 20}]
(* Eric W. Weisstein, Sep 18 2017 *)

a(n) = n!*(n-1) * (n-1 + Sum_{k=1..floor(3*(n-1)/2)} k*A007799(n, k)). - Andrew Howroyd, May 13 2017

a(7)-a(16) from Andrew Howroyd, May 13 2017

A284039 Wiener index of the n-permutation star graph.

Original entry on oeis.org

0, 1, 27, 744, 26520, 1239840, 74662560, 5663831040, 530098007040, 60105991680000, 8127440487936000, 1292894601191424000, 239129895342514176000, 50899158690744139776000, 12356174324714508288000000, 3393918280427832764006400000, 1047355019625604129593753600000

Offset: 1

Eric W. Weisstein, Sep 13 2017

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..50
Eric Weisstein's World of Mathematics, Permutation Star Graph
Eric Weisstein's World of Mathematics, Wiener Index

Cf. A007799, A192837.

a[n_, 0] = 1;
a[n_, 1] = n - 1;
a[n_, 2] = (n - 1) (n - 2);
a[n_, k_ /; k >= 2] := a[n, k] = (n - 1) a[n - 1, k - 1] + Sum[j a[j, k - 3], {j, 1, n - 2}];
Table[n!/2 Sum[k a[n, k], {k, Floor[3 (n - 1)/2]}], {n, 10}]

a(n) = n!/2 * Sum_{k=1..floor(3*(n-1)/2)} k*A007799(n, k). - Andrew Howroyd, Sep 17 2017

Terms a(8) and beyond from Andrew Howroyd, Sep 17 2017

A296057 Denominators of Harary index for the n-permutation star graph.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 11, 143, 13, 13, 17, 19, 323, 323, 437, 437, 23, 1, 667, 20677, 899, 31, 899, 33263, 33263, 33263, 47027, 2022161, 65231, 65231, 3065857, 3065857, 3065857, 3065857, 4391633, 4391633, 4391633, 4391633, 6319667, 385499687, 8965109, 8965109

Offset: 1

Andrew Howroyd, Dec 09 2017

Eric Weisstein's World of Mathematics, Harary Index
Eric Weisstein's World of Mathematics, Permutation Star Graph

Cf. A296190 (numerators), A007799, A284039.

A007799[n_, i_] := Sum[Binomial[n - 1, k] Binomial[n - 1 - k, t] StirlingS1[k + 1, i - k + 1 - 2 t] (-1)^(i + 2 - t), {k, 0, Min[n - 1, i + 1]}, {t, Max[0, Ceiling[(i - 2 k)/2]], Min[n - 1 - k, Floor[(i + 1 - k)/2]]}];
Table[n! Sum[A007799[n, k]/k, {k, Floor[3 (n - 1)/2]}]/2, {n, 20}] // Denominator (* Eric W. Weisstein, Dec 09 2017 *)

A296190 Numerators of Harary index for the n-permutation star graph.

Original entry on oeis.org

0, 1, 10, 123, 2202, 59040, 2287680, 121394000, 92649740400, 105538103163360, 1034297134668000, 134399089883282400, 27076064087538702720, 5451799851068349018240, 19300076847195336557164800, 4599598343095846092562560000, 1682634821690958905899793664000

Offset: 1

Eric W. Weisstein, Dec 07 2017

Eric Weisstein's World of Mathematics, Harary Index
Eric Weisstein's World of Mathematics, Permutation Star Graph

Cf. A296057 (denominators), A007799, A284039.

A007799[n_, i_] := Sum[Binomial[n - 1, k] Binomial[n - 1 - k, t] StirlingS1[k + 1, i - k + 1 - 2 t] (-1)^(i + 2 - t), {k, 0, Min[n - 1, i + 1]}, {t, Max[0, Ceiling[(i - 2 k)/2]], Min[n - 1 - k, Floor[(i + 1 - k)/2]]}];
Table[n! Sum[A007799[n, k]/k, {k, Floor[3 (n - 1)/2]}]/2, {n, 20}] // Numerator (* Eric W. Weisstein, Dec 09 2017 *)

a(n)/A296057(n) = (n!/2) * Sum_{k=1..floor(3*(n-1)/2)} A007799(n, k)/k. - Andrew Howroyd, Dec 09 2017

a(9)-a(17) from Andrew Howroyd, Dec 09 2017

cp's OEIS Frontend

A004070 Table of Whitney numbers W(n,k) read by antidiagonals, where W(n,k) is maximal number of pieces into which n-space is sliced by k hyperplanes, n >= 0, k >= 0.

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A192837 Molecular topological indices of the permutation star graphs.

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A284039 Wiener index of the n-permutation star graph.

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A296057 Denominators of Harary index for the n-permutation star graph.

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A296190 Numerators of Harary index for the n-permutation star graph.

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