A143858 -id:A143858 - cp's OEIS frontend

A258993 Triangle read by rows: T(n,k) = binomial(n+k,n-k), k = 0..n-1.

1, 1, 3, 1, 6, 5, 1, 10, 15, 7, 1, 15, 35, 28, 9, 1, 21, 70, 84, 45, 11, 1, 28, 126, 210, 165, 66, 13, 1, 36, 210, 462, 495, 286, 91, 15, 1, 45, 330, 924, 1287, 1001, 455, 120, 17, 1, 55, 495, 1716, 3003, 3003, 1820, 680, 153, 19, 1, 66, 715, 3003, 6435, 8008, 6188, 3060, 969, 190, 21

Offset: 1

Reinhard Zumkeller, Jun 22 2015

T(n,k) = A085478(n,k) = A007318(A094727(n),A004736(k)), k = 0..n-1;
rounded(T(n,k)/(2*k+1)) = A258708(n,k);
rounded(sum(T(n,k)/(2*k+1)): k = 0..n-1) = A000967(n).

			.  n\k |  0  1    2    3     4     5     6     7    8    9  10 11
. -----+-----------------------------------------------------------
.   1  |  1
.   2  |  1  3
.   3  |  1  6    5
.   4  |  1 10   15    7
.   5  |  1 15   35   28     9
.   6  |  1 21   70   84    45    11
.   7  |  1 28  126  210   165    66    13
.   8  |  1 36  210  462   495   286    91    15
.   9  |  1 45  330  924  1287  1001   455   120   17
.  10  |  1 55  495 1716  3003  3003  1820   680  153   19
.  11  |  1 66  715 3003  6435  8008  6188  3060  969  190  21
.  12  |  1 78 1001 5005 12870 19448 18564 11628 4845 1330 231 23  .

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

If a diagonal of 1's is added on the right, this becomes A085478.
Essentially the same as A143858.
Cf. A007318, A004736, A094727.
Cf. A027941 (row sums), A117671 (central terms), A143858, A000967, A258708.
T(n,k): A000217 (k=1), A000332 (k=2), A000579 (k=3), A000581 (k=4), A001287 (k=5), A010965 (k=6), A010967 (k=7), A010969 (k=8), A010971 (k=9), A010973 (k=10), A010975 (k=11), A010977 (k=12), A010979 (k=13), A010981 (k=14), A010983 (k=15), A010985 (k=16), A010987 (k=17), A010989 (k=18), A010991 (k=19), A010993 (k=20), A010995 (k=21), A010997 (k=22), A010999 (k=23), A011001 (k=24), A017714 (k=25), A017716 (k=26), A017718 (k=27), A017720 (k=28), A017722 (k=29), A017724 (k=30), A017726 (k=31), A017728 (k=32), A017730 (k=33), A017732 (k=34), A017734 (k=35), A017736 (k=36), A017738 (k=37), A017740 (k=38), A017742 (k=39), A017744 (k=40), A017746 (k=41), A017748 (k=42), A017750 (k=43), A017752 (k=44), A017754 (k=45), A017756 (k=46), A017758 (k=47), A017760 (k=48), A017762 (k=49), A017764 (k=50).
T(n+k,n): A005408 (k=1), A000384 (k=2), A000447 (k=3), A053134 (k=4), A002299 (k=5), A053135 (k=6), A053136 (k=7), A053137 (k=8), A053138 (k=9), A196789 (k=10).
Cf. A165253.

Flat(List([1..12], n-> List([0..n-1], k-> Binomial(n+k,n-k) ))); # G. C. Greubel, Aug 01 2019

a258993 n k = a258993_tabl !! (n-1) !! k
a258993_row n = a258993_tabl !! (n-1)
a258993_tabl = zipWith (zipWith a007318) a094727_tabl a004736_tabl

[Binomial(n+k,n-k): k in [0..n-1], n in [1..12]]; // G. C. Greubel, Aug 01 2019

Table[Binomial[n+k,n-k], {n,1,12}, {k,0,n-1}]//Flatten (* G. C. Greubel, Aug 01 2019 *)

T(n,k) = binomial(n+k,n-k);
for(n=1, 12, for(k=0,n-1, print1(T(n,k), ", "))) \\ G. C. Greubel, Aug 01 2019

[[binomial(n+k,n-k) for k in (0..n-1)] for n in (1..12)] # G. C. Greubel, Aug 01 2019

T(n,k) = A085478(n,k) = A007318(A094727(n),A004736(k)), k = 0..n-1;
rounded(T(n,k)/(2*k+1)) = A258708(n,k);
rounded(sum(T(n,k)/(2*k+1)): k = 0..n-1) = A000967(n).

A284938 Triangle read by rows: coefficients of the edge cover polynomial for the n-path graph P_n.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 3, 4, 1, 0, 0, 0, 0, 1, 6, 5, 1, 0, 0, 0, 0, 0, 4, 10, 6, 1, 0, 0, 0, 0, 0, 1, 10, 15, 7, 1, 0, 0, 0, 0, 0, 0, 5, 20, 21, 8, 1, 0, 0, 0, 0, 0, 0, 1, 15, 35, 28, 9, 1, 0, 0, 0, 0, 0, 0, 0, 6, 35, 56, 36, 10, 1, 0, 0, 0, 0, 0, 0, 0, 1, 21, 70, 84, 45, 11, 1

Offset: 1

Eric W. Weisstein, Apr 06 2017

			0;
0,1;
0,0,1;
0,0,1,1;
0,0,0,2,1;
0,0,0,1,3,1;
0,0,0,0,3,4,1;
0,0,0,0,1,6,5,1;
0,0,0,0,0,4,10,6,1;
0,0,0,0,0,1,10,15,7,1;
0,0,0,0,0,0,5,20,21,8,1;
0,0,0,0,0,0,1,15,35,28,9,1;
0,0,0,0,0,0,0,6,35,56,36,10,1;
0,0,0,0,0,0,0,1,21,70,84,45,11,1;
...

Eric Weisstein's World of Mathematics, Edge Cover Polynomial
Eric Weisstein's World of Mathematics, Path Graph

Unsigned version of A057094.
Row sums are A000045(n-1).
Cf. A286912, A258993, A030528, A085478, A098925, A102426, A143858, etc.

Prepend[CoefficientList[Table[x^(n/2) Fibonacci[n - 1, Sqrt[x]], {n, 2, 14}], x], {0}] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
Prepend[CoefficientList[LinearRecurrence[{x, x}, {0, x}, {2, 14}], x], {0}] // Flatten (* Eric W. Weisstein, Apr 07 2017 *)

a(n) = abs(A057094(n)).

A000967 Sum of Fermat coefficients.

Original entry on oeis.org

1, 2, 4, 8, 18, 40, 91, 210, 492, 1165, 2786, 6710, 16267, 39650, 97108, 238824, 589521, 1459960, 3626213, 9030450, 22542396, 56393792, 141358274, 354975429, 892893120, 2249412290, 5674891000, 14335757256, 36259245522, 91815545800

Offset: 1

N. J. A. Sloane

nonn

			n...Sum_{c=1..n} (n:c).....a(n)
--------------------------------
.1........1.................1
.2........2.................2
.3........4.................4
.4........8+1/3.............8

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).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000 (corrected by Sean A. Irvine, April 17, 2019)
R. P. Loh, A. G. Shannon, A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
P. A. Piza, Fermat coefficients, Math. Mag., 27 (1954), 141-146.

Cf. A143858, A258708.

import Data.Function (on)
a000967 n = round $ sum $
            zipWith ((/) `on` fromIntegral) (a258993_row n) [1, 3 ..]
-- Reinhard Zumkeller, Jun 22 2015

[Round((&+[Binomial(n+k,n-k)/(2*k+1): k in [0..n-1]])): n in [1..35]]; // G. C. Greubel, Apr 16 2019

FermatCoeff:=(n,c)->binomial(2*n-c,c-1)/c:seq(round(add(FermatCoeff(n,c),c=1..n)),n=1..40); # Pab Ter, Oct 13 2005

Table[Round[Sum[Binomial[n+k, n-k]/(2*k+1), {k, 0, n-1}]], {n,1,35}] (* G. C. Greubel, Apr 16 2019 *)

{a(n) = round(sum(k=0,n-1, binomial(n+k,n-k)/(2*k+1)))}; \\ G. C. Greubel, Apr 16 2019

[round(sum(binomial(n+k,n-k)/(2*k+1) for k in (0..n-1))) for n in (1..35)] # G. C. Greubel, Apr 16 2019

Following Piza's definition for the Fermat coefficients: (n:c)=binomial(2n-c, c-1)/c, a(n)= Round( sum_ {c=1..n} (n:c) ). - Pab Ter (pabrlos2(AT)yahoo.com), Oct 13 2005

a(n) = rounded(sum(A258993(n,k)/(2*k+1)): k = 0..n-1). - Reinhard Zumkeller, Jun 22 2015

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 13 2005

A123970 Triangle read by rows: T(0,0)=1; T(n,k) is the coefficient of x^(n-k) in the monic characteristic polynomial of the n X n matrix (min(i,j)) (i,j=1,2,...,n) (0 <= k <= n, n >= 1).

Original entry on oeis.org

1, 1, -1, 1, -3, 1, 1, -6, 5, -1, 1, -10, 15, -7, 1, 1, -15, 35, -28, 9, -1, 1, -21, 70, -84, 45, -11, 1, 1, -28, 126, -210, 165, -66, 13, -1, 1, -36, 210, -462, 495, -286, 91, -15, 1, 1, -45, 330, -924, 1287, -1001, 455, -120, 17, -1, 1, -55, 495, -1716, 3003, -3003, 1820, -680, 153, -19, 1, 1, -66, 715, -3003, 6435, -8008

Offset: 0

Gary W. Adamson and Roger L. Bagula, Oct 29 2006

This sequence is the same as A129818 up to sign. - T. D. Noe, Sep 30 2011

Riordan array (1/(1-x), -x/(1-x)^2). - Philippe Deléham, Feb 18 2012

			Triangular sequence (gives the odd Tutte-Beraha constants as roots!) begins:
  1;
  1,  -1;
  1,  -3,   1;
  1,  -6,   5,   -1;
  1, -10,  15,   -7,    1;
  1, -15,  35,  -28,    9,    -1;
  1, -21,  70,  -84,   45,   -11,   1;
  1, -28, 126, -210,  165,   -66,  13,   -1;
  1, -36, 210, -462,  495,  -286,  91,  -15,  1;
  1, -45, 330, -924, 1287, -1001, 455, -120, 17, -1;
  ...

S. Beraha, Infinite non-trivial families of maps and chromials, Ph.D. thesis. Baltimore, MD: Johns Hopkins University, 1975.
Steven R. Finch, Mathematical Constants (Encyclopedia of Mathematics and its Applications), chapter 5.25.
W. T. Tutte, "More about Chromatic Polynomials and the Golden Ratio." In Combinatorial Structures and their Applications: Proc. Calgary Internat. Conf., Calgary, Alberta, 1969. New York: Gordon and Breach, p. 439, 1969.

Paul Fendley and Vyacheslav Krushkal, Tutte chromatic identities and the Temperley-Lieb algebra, arXiv:0711.0016 [math.CO], 2007-2008.
Donatella Merlini and Renzo Sprugnoli, Arithmetic into geometric progressions through Riordan arrays, Discrete Mathematics 340.2 (2017): 160-174. See page 161.
Eric Weisstein's World of Mathematics, Beraha Constants

Cf. A085478, A076756.
Cf. A109954, A129818, A143858, A165253. - R. J. Mathar, Jan 10 2011
Modulo signs, inverse matrix to A039599.

/* As triangle */ [[(-1)^k*Binomial(n + k, 2*k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jan 04 2019

with(linalg): m:=(i,j)->min(i,j): M:=n->matrix(n,n,m): T:=(n,k)->coeff(charpoly(M(n),x),x,n-k): 1; for n from 1 to 11 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

An[d_] := MatrixPower[Table[Min[n, m], {n, 1, d}, {m, 1, d}], -1]; Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[An[d], x], x], {d, 1, 20}]]; Flatten[%]

f(n,x) = (2x-1)f(n-1,x)-x^2*f(n-2,x), where f(n,x) is the characteristic polynomial of the n X n matrix from the definition and f(0,x)=1. See formula in Fendley and Krushkal. - Jonathan Vos Post, Nov 04 2007
T(n,k) = (-1)^k * A085478(n,k) = (-1)^n * A129818(n,k). - Philippe Deléham, Feb 06 2012
T(n,k) = 2*T(n-1,k) - T(n-1,k-1) - T(n-2,k), T(0,0)=T(1,0)=1, T(1,1)=-1, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Nov 29 2013

Edited by N. J. A. Sloane, Nov 29 2006

A258935 Independence number of Keller graphs.

Original entry on oeis.org

4, 5, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset: 1

Stan Wagon, Nov 06 2015

			For G(2), a maximum independent set is {03,10,12,13,23}.

W. Jarnicki, W. Myrvold, P. Saltzman, S. Wagon, Properties, proved and conjectured, of Keller, queen, and Mycielski graphs, Ars Mathematica Contemporanea 13:2 (2017) 427-460.

Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Eric Weisstein's World of Mathematics, Independence Number
Eric Weisstein's World of Mathematics, Keller Graph

Cf. A006946, A135831, A135907.

Essentially the same as A143858, A240951, A198633, A171497, A151821, A146541 and A077552.

Mathematica
```
Join[{4, 5}, 2^Range[3, 10]]
```

a(n)=if(n>2,2^n,n+3) \\ Charles R Greathouse IV, Nov 07 2015

a(n) = 2^n except a(1) = 4 and a(2) = 5.

G.f.: x*(x*(3+2*x)-4)/(2*x-1), e.g.f.: exp(2*x)+x^2/2+2*x-1. - Benedict W. J. Irwin, Jul 15 2016

cp's OEIS Frontend

A258993 Triangle read by rows: T(n,k) = binomial(n+k,n-k), k = 0..n-1.

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A284938 Triangle read by rows: coefficients of the edge cover polynomial for the n-path graph P_n.

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A000967 Sum of Fermat coefficients.

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A123970 Triangle read by rows: T(0,0)=1; T(n,k) is the coefficient of x^(n-k) in the monic characteristic polynomial of the n X n matrix (min(i,j)) (i,j=1,2,...,n) (0 <= k <= n, n >= 1).

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Crossrefs

Programs

Magma

Maple

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Extensions

A258935 Independence number of Keller graphs.

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Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula