A117317 -id:A117317 - cp's OEIS frontend

A185342 Triangle of successive recurrences in columns of A117317(n).

2, 4, -4, 6, -12, 8, 8, -24, 32, -16, 10, -40, 80, -80, 32, 12, -60, 160, -240, 192, -64, 14, -84, 280, -560, 672, -448, 128, 16, -112, 448, -1120, 1792, -1792, 1024, -256, 18, -144, 672, -2016, 4032, -5376, 4608, -2304, 512, 20, -180, 960, -3360, 8064

Offset: 0

Paul Curtz, Jan 26 2012

A117317 (A):
1
2    1
4    5   1
8   16   9   1
16  44  41  14   1
32 112 146  85  20  1
64 272 456 377 155 27 1
have for their columns successive signatures
(2) (4,-4) (6,-12,8) (8,-24, 32, -16) (10,-40,80,-80,32) i.e. a(n).
Take based on abs(A133156) (B):
1
2   0
4   1  0
8   4  0 0
16 12  1 0 0
32 32  6 0 0 0
64 80 24 1 0 0 0.
The recurrences of successive columns are also a(n). a(n) columns: A005843(n+1), A046092(n+1), A130809, A130810, A130811, A130812, A130813.
A053220 + A001787 = A014480.

			Triangle T(n,k),for 1<=k<=n, begins :
2                                         (1)
4    -4                                   (2)
6   -12   8                               (3)
8   -24  32   -16                         (4)
10  -40  80   -80   32                    (5)
12  -60 160  -240  192   -64              (6)
14  -84 280  -560  672  -448  128         (7)
16 -112 448 -1120 1792 -1792 1024 -256    (8)
Successive rows can be divided by A171977.

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. For (A): A053220, A056243. For (B): A000079, A001787, A001788, A001789. For A193862: A115068 (a Coxeter group). For (2): A014480 (also (3),(4),(5),..); also A053220 and A001787.

Cf. A007318.

Table[(-1)*Binomial[n, k]*(-2)^k, {n, 1, 20}, {k, 1, n}] // Flatten (* G. C. Greubel, Jun 27 2017 *)

for(n=1,20, for(k=1,n, print1((-2)^(k+1)*binomial(n,k)/2, ", "))) \\ G. C. Greubel, Jun 27 2017

Take A133156(n) without 1's or -1's double triangle (C)=
2
4
8      -4
16    -12
32    -32   6
64    -80  24
128  -192  80   -8
256  -448 240  -40
512 -1024 672 -160 10;
a(n) is increasing odd diagonals and increasing (sign changed) even diagonals. Rows sum of (C) = A201629 (?) Another link between Chebyshev polynomials and cos( ).
Absolute values: A013609(n) without 1's. Also 2*A193862 = (2*A002260)*A135278.
T(n,k) = T(n-1,k) - 2*T(n-1,k-1) for k>1, T(n,1) = 2*n = 2*T(n-1,1) - T(n-2,1). - Philippe Deléham, Feb 11 2012
T(n,k) = (-1)* Binomial(n,k)*(-2)^k, 1<=k<=n. - Philippe Deléham, Feb 11 2012

A056242 Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).

Original entry on oeis.org

1, 1, 2, 1, 5, 4, 1, 9, 16, 8, 1, 14, 41, 44, 16, 1, 20, 85, 146, 112, 32, 1, 27, 155, 377, 456, 272, 64, 1, 35, 259, 833, 1408, 1312, 640, 128, 1, 44, 406, 1652, 3649, 4712, 3568, 1472, 256, 1, 54, 606, 3024, 8361, 14002, 14608, 9312, 3328, 512, 1, 65, 870, 5202

Offset: 1

Colin Mallows, Aug 23 2000

Generalized Riordan array (1/(1-x), x/(1-x) + x*dif(x/1-x),x)). - Paul Barry, Dec 26 2007
Reversal of A117317. - Philippe Deléham, Feb 11 2012
Essentially given by (1, 0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 11 2012
This sequence is given in the Strehl presentation with the o.g.f. (1-z)/[1-2(1+t)z+(1+t)z^2], with offset 0, along with a recursion relation, a combinatorial interpretation, and relations to Hermite and Laguerre polynomials. Note that the o.g.f. is related to that of A049310. - Tom Copeland, Jan 08 2017
From Gus Wiseman, Mar 06 2020: (Start)
T(n,k) is also the number of unimodal length-n sequences covering an initial interval of positive integers with maximum part k, where a sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence. For example, the sequences counted by row n = 4 are:
  (1111)  (1112)  (1123)  (1234)
          (1121)  (1132)  (1243)
          (1122)  (1223)  (1342)
          (1211)  (1231)  (1432)
          (1221)  (1232)  (2341)
          (1222)  (1233)  (2431)
          (2111)  (1321)  (3421)
          (2211)  (1322)  (4321)
          (2221)  (1332)
                  (2231)
                  (2311)
                  (2321)
                  (2331)
                  (3211)
                  (3221)
                  (3321)
(End)
T(n,k) is the number of hexagonal directed-column convex polyominoes of area n with k columns (see Baril et al. at page 9). - Stefano Spezia, Oct 14 2023

			Triangle begins:
  1;
  1,    2;
  1,    5,    4;
  1,    9,   16,    8;
  1,   14,   41,   44,   16;
  1,   20,   85,  146,  112,   32;
  1,   27,  155,  377,  456,  272,   64;
  1,   35,  259,  833, 1408, 1312,  640,  128;
  1,   44,  406, 1652, 3649, 4712, 3568, 1472,  256;
T(3,2)=5 because we have {1}{23}, {23}{1}, {12}{3}, {3}{12} and {2}{13}.
Triangle (1, 0, 1/2, 1/2, 0, 0, 0, ...) DELTA (0, 2, 0, 0, 0, ...) begins:
  1;
  1,   0;
  1,   2,   0;
  1,   5,   4,   0;
  1,   9,  16,   8,   0;
  1,  14,  41,  44,  16,   0;
  1,  20,  85, 146, 112,  32,   0;
  1,  27, 155, 377, 456, 272,  64,   0;

Reinhard Zumkeller, Rows n = 1..125 of table, flattened
Jean-Luc Baril, José L. Ramírez, and Fabio A. Velandia, Bijections between Directed-Column Convex Polyominoes and Restricted Compositions, September 29, 2023.
Tyler Clark and Tom Richmond, The Number of Convex Topologies on a Finite Totally Ordered Set, 2013, Involve, Vol. 8 (2015), No. 1, 25-32.
F. K. Hwang and C. L. Mallows, Enumerating nested and consecutive partitions, J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.
Finn Bjarne Jost, Tautological Intersection Numbers and Order-Consecutive Partition Sequences, arXiv:2307.15825 [math.CO], 2023. See p. 9.
V. Strehl, Combinatoire rétrospective et créative, an on-line presentation, slide 36, SLC 71, Bertinoro,, September 18, 2013.
Volker Strehl, Lacunary Laguerre Series from a Combinatorial Perspective, Séminaire Lotharingien de Combinatoire, B76c (2017).

Row sums are A007052.
Column k = n - 1 is A053220.
Ordered set-partitions are A000670.
Cf. A001523, A049310, A072704, A084938, A097805, A117317, A227038, A328509, A332294, A332673, A332724, A332872.

a056242 n k = a056242_tabl !! (n-1)!! (k-1)
a056242_row n = a056242_tabl !! (n-1)
a056242_tabl = [1] : [1,2] : f [1] [1,2] where
   f us vs = ws : f vs ws where
     ws = zipWith (-) (map (* 2) $ zipWith (+) ([0] ++ vs) (vs ++ [0]))
                      (zipWith (+) ([0] ++ us ++ [0]) (us ++ [0,0]))
-- Reinhard Zumkeller, May 08 2014

T:=proc(n,k) if k=1 then 1 elif k<=n then sum((-1)^(k-1-j)*binomial(k-1,j)*binomial(n+2*j-1,2*j),j=0..k-1) else 0 fi end: seq(seq(T(n,k),k=1..n),n=1..12);

rows = 11; t[n_, k_] := (-1)^(k+1)*HypergeometricPFQ[{1-k, (n+1)/2, n/2}, {1/2, 1}, 1]; Flatten[ Table[ t[n, k], {n, 1, rows}, {k, 1, n}]](* Jean-François Alcover, Nov 17 2011 *)

The Hwang and Mallows reference gives explicit formulas.
T(n,k) = Sum_{j=0..k-1} (-1)^(k-1-j)*binomial(k-1, j)*binomial(n+2j-1, 2j) (1<=k<=n); this is formula (11) in the Huang and Mallows reference.
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), T(1,1) = 1, T(2,1) = 1, T(2,2) = 2. - Philippe Deléham, Feb 11 2012
G.f.: -(-1+x)*x*y/(1-2*x-2*x*y+x^2*y+x^2). - R. J. Mathar, Aug 11 2015

A086405 Row T(n,3) of number array A086404.

Original entry on oeis.org

1, 4, 18, 84, 396, 1872, 8856, 41904, 198288, 938304, 4440096, 21010752, 99423936, 470479104, 2226331008, 10535111424, 49852682496, 235905426432, 1116316463616, 5282466223104, 24996898556928, 118286594002944

Offset: 0

Paul Barry, Jul 19 2003

Binomial transform of A079935.

Number of nonisomorphic graded posets with 0 of rank n+1, with exactly 2 elements of each rank level above 0. Here, we do not assume all maximal elements have maximal rank and thus use graded poset to mean: For every element x, all maximal chains among those with x as greatest element have the same finite length. - David Nacin, Feb 13 2012

R. Stanley, Enumerative combinatorics, Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wikipedia, Graded Poset
Index entries for linear recurrences with constant coefficients, signature (6,-6).

Cf. A079935, A086404.

LinearRecurrence[{6, -6}, {1, 4}, 60] (* David Nacin, Feb 27 2012 *)

def a(n, adict={0:1, 1:4}):
    if n in adict:
        return adict[n]
    adict[n]=6*a(n-1)-6*a(n-2)
    return adict[n] # David Nacin, Feb 27 2012

G.f.: (1-2*x)/((1-(3-sqrt(3))*x)*(1-(3+sqrt(3))*x)) = (1-2*x)/(1-6*x+6*x^2);
a(n) = (3-sqrt(3))^n*(1/2 - 1/(2*sqrt(3))) + (3 + sqrt(3))^n*(1/2 + 1/(2*sqrt(3))).
E.g.f.: exp(3*x)*(cosh(sqrt(3)*x) + sinh(sqrt(3)*x)/sqrt(3)). - Paul Barry, Nov 20 2003
a(n) = Sum_{k=1..floor(n/2)} C(n, 2k)*3^(n-k-1). - Paul Barry, Nov 22 2003
a(n) = (((1+sqrt(3))*(3+sqrt(3))^n) - ((1-sqrt(3))*(3-sqrt(3))^n))/sqrt(12). - Al Hakanson (hawkuu(AT)gmail.com), Jun 10 2009
a(n) = Sum_{k=0..n} A117317(n,k)*2^k. - Philippe Deléham, Jan 28 2012
a(n) = 6*(a(n-1) - a(n-2)), a(0)=1, a(1)=4. - David Nacin, Feb 27 2012
G.f.: (1-2*x)/(1-6*x+6*x^2). - Colin Barker, Aug 04 2012

A090040 (3*6^n + 2^n)/4.

Original entry on oeis.org

1, 5, 28, 164, 976, 5840, 35008, 209984, 1259776, 7558400, 45349888, 272098304, 1632587776, 9795522560, 58773127168, 352638746624, 2115832446976, 12694994616320, 76169967566848, 457019805138944, 2742118830309376

Offset: 0

Paul Barry, Nov 20 2003

A090040 is the Q-residue of the triangle A175840, where Q is the triangular array (t(i,j)) given by t(i,j)=1; see A193649 for the definition of Q-residue. - Clark Kimberling, Aug 07 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (8,-12).

Cf. A081335.

[(3*6^n+2^n)/4: n in [0..30]]; // Vincenzo Librandi, Jun 10 2011

LinearRecurrence[{8,-12},{1,5},30] (* Harvey P. Dale, Nov 23 2014 *)

a(n)=(3*6^n+2^n)/4 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (1-3*x)/((1-2*x)*(1-6*x)).
E.g.f.: (3*exp(6*x)+exp(2*x))/4 = exp(4*x)*(cosh(2*x)+sinh(2*x)/2).
a(n) = 8*a(n-1) -12*a(n-2), a(0)=1, a(1)=5.
a(n) = (3*6^n+2^n)/4.
a(n)=6*a(n-1)-2^(n-1). - Paul Curtz, Jan 09 2009
Fourth binomial transform of (1, 1, 4, 4, 16, 16, ...). a(n)=sum{k=1..floor(n/2), C(n, 2k)4^(n-k-1)}. - Paul Barry, Nov 22 2003
a(n) = A019590 (mod 4), proof via a(n)=8*a(n-1)-12*a(n-2). - R. J. Mathar, Feb 25 2009
a(n) = Sum_{k, 0<=k<=n} A117317(n,k)*3^k. - Philippe Deléham, Jan 28 2012

A165241 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,1,0,0,0,0,0,0,0,...] DELTA [1,0,1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 4, 9, 6, 1, 8, 24, 25, 10, 1, 16, 60, 85, 55, 15, 1, 32, 144, 258, 231, 105, 21, 1, 64, 336, 728, 833, 532, 182, 28, 1, 128, 768, 1952, 2720, 2241, 1092, 294, 36, 1, 256, 1728, 5040, 8280, 8361, 5301, 2058, 450, 45, 1

Offset: 0

Philippe Deléham, Sep 09 2009

Rows sums: A006012; Diagonal sums: A052960.

The sums of each column of A117317 with its subsequent column, treated as a lower triangular matrix with an initial null column attached, or, equivalently, the products of the row polynomials p(n,y) of A117317 with (1+y) with the initial first row below added to the final result. The reversal of A117317 is A056242 with several combinatorial interpretations. - Tom Copeland, Jan 08 2017

			Triangle begins:
  1;
  1,  1;
  2,  3,  1;
  4,  9,  6,  1;
  8, 24, 25, 10,  1; ...

Cf. A000012, A000217, A005582, A011782, A084858.

Cf. A056242, A117317.

Sum_{k=0..n} T(n,k)*x^k = A009116(n), A000007(n), A011782(n), A006012(n), A083881(n), A081335(n), A090139(n), A145301(n), A145302(n), A145303(n), A143079(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively. Sum_{k=0..n} T(n,k)*x^(n-k) = A123335(n), A000007(n), A000012(n), A006012(n), A084120(n), A090965(n), A165225(n), A165229(n), A165230(n), A165231(n), A165232(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively.
G.f.: (1-(1+y)*x)/(1-2(1+y)*x+(y+y^2)*x^2). - Philippe Deléham, Dec 19 2011
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2) with T(0,0) = T(1,0) = T(1,1) = 1 and T(n,k) = 0 if k<0 or if nPhilippe Deléham, Dec 19 2011

O.g.f. corrected by Tom Copeland, Jan 15 2017

A090041 a(n) = 10a(n-1) - 20a(n-2), a(0)=1, a(1)=6.

Original entry on oeis.org

1, 6, 40, 280, 2000, 14400, 104000, 752000, 5440000, 39360000, 284800000, 2060800000, 14912000000, 107904000000, 780800000000, 5649920000000, 40883200000000, 295833600000000, 2140672000000000, 15490048000000000

Offset: 0

Paul Barry, Nov 20 2003

Index entries for linear recurrences with constant coefficients, signature (10,-20).

Cf. A090139, A117317.

G.f.: (1-4*x)/(1-10*x+20*x^2) = (1-4*x)/((1-(5-sqrt(5))*x)*(1-(5+sqrt(5))*x)).
E.g.f.: exp(5*x)*(cosh(sqrt(5)*x) + sinh(sqrt(5)*x)/sqrt(5));
a(n) = ((1+sqrt(5))*(5+sqrt(5))^n - (1-sqrt(5))*(5-sqrt(5))^n)/(2*sqrt(5)).
Fifth binomial transform of (1, 1, 5, 5, 25, 25, ...). - Paul Barry, Nov 22 2003
3rd binomial transform of Fibonacci(3n+1). - Paul Barry, Mar 23 2004
a(n) = Sum_{k=0..n} A117317(n,k)*4^k. - Philippe Deléham, Jan 28 2012

cp's OEIS Frontend

A185342 Triangle of successive recurrences in columns of A117317(n).

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A056242 Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).

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A086405 Row T(n,3) of number array A086404.

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A090040 (3*6^n + 2^n)/4.

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A165241 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,1,0,0,0,0,0,0,0,...] DELTA [1,0,1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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A090041 a(n) = 10*a(n-1) - 20*a(n-2), a(0)=1, a(1)=6.

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A090041 a(n) = 10a(n-1) - 20a(n-2), a(0)=1, a(1)=6.