A165253 -id:A165253 - 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).

A121314 Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 5, 6, 1, 0, 1, 7, 15, 10, 1, 0, 1, 9, 28, 35, 15, 1, 0, 1, 11, 45, 84, 70, 21, 1, 0, 1, 13, 66, 165, 210, 126, 28, 1, 0, 1, 15, 91, 286, 495, 462, 210, 36, 1

Offset: 0

Philippe Deléham, Aug 25 2006

A054142 with first diagonal 1, 0, 0, 0, 0, 0, 0, 0, ...

Mirror image of triangle in A165253.

			Triangle begins
  1;
  0,  1;
  0,  1,  1;
  0,  1,  3,  1;
  0,  1,  5,  6,  1;
  0,  1,  7, 15, 10,  1;
  0,  1,  9, 28, 35, 15,  1;
  0,  1, 11, 45, 84, 70, 21,  1;

Alois P. Heinz, Rows n = 0..140, flattened
F. Yano and H. Yoshida, Some set partition statistics in non-crossing partitions and generating functions, Discr. Math., 307 (2007), 3147-3160.

Cf. A054142, A165253.

T(0,0)=1; T(n,0)=0 for n > 0; T(n+1,k+1) = binomial(2*n-k,k)for n >= 0 and k >= 0.
Sum_{k=0..n} T(n,k)*x^k = A001519(n), A047849(n), A165310(n), A165311(n), A165312(n), A165314(n), A165322(n), A165323(n), A165324(n) for x = 1,2,3,4,5,6,7,8,9 respectively.
Sum_{k=0..n} 2^k*T(n,k) = (4^n+2)/3.
Sum_{k=0..n} 2^(n-k)*T(n,k) = A001835(n).
Sum_{k=0..n} 3^k*4^(n-k)*T(n,k) = A054879(n). - Philippe Deléham, Aug 26 2006
Sum_{k=0..n} T(n,k)*(-1)^k*2^(3n-2k) = A143126(n). - Philippe Deléham, Oct 31 2008
Sum_{k=0..n} T(n,k)*(-1)^k*3^(n-k) = A138340(n)/4^n. - Philippe Deléham, Nov 01 2008
G.f.: (1-(y+1)*x)/(1-(2y+1)*x+y^2*x^2). - Philippe Deléham, Nov 01 2011
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-2), T(0,0) = T(1,1) = 1, T(1,0) = 0. - Philippe Deléham, Feb 19 2012

A165310 a(0)=1, a(1)=3, a(n) = 7a(n-1) - 9a(n-2) for n > 1.

Original entry on oeis.org

1, 3, 12, 57, 291, 1524, 8049, 42627, 225948, 1197993, 6352419, 33684996, 178623201, 947197443, 5022773292, 26634636057, 141237492771, 748950724884, 3971517639249, 21060066950787, 111676809902268, 592197066758793

Offset: 0

Philippe Deléham, Sep 14 2009

nonn

a(n)/a(n-1) tends to (7+sqrt(13))/2 = 5.30277563... = 2+A098316.

For n >= 2, a(n) equals 3^n times the permanent of the (2n-2) X (2n-2) matrix with 1/sqrt(3)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (7,-9).

Cf. A098316, A165253.

I:=[1,3]; [n le 2 select I[n] else 7*Self(n-1)-9*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 24 2011

LinearRecurrence[{7,-9},{1,3},30] (* Harvey P. Dale, Sep 23 2011 *)

G.f.: (1-4x)/(1-7x+9x^2).
a(n) = Sum_{k=0..n} A165253(n,k)*3^(n-k).
a(n) = ((13-sqrt(13))*(7+sqrt(13))^n+(13+sqrt(13))*(7-sqrt(13))^n )/(26*2^n). - Klaus Brockhaus, Sep 26 2009

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

A165312 a(0)=1, a(1)=5, a(n)=11a(n-1)-25a(n-2) for n>1.

Original entry on oeis.org

1, 5, 30, 205, 1505, 11430, 88105, 683405, 5314830, 41378005, 322287305, 2510710230, 19560629905, 152399173205, 1187375157630, 9251147403805, 72078242501105, 561581982417030, 4375445744059705, 34090353624231005

Offset: 0

Philippe Deléham, Sep 14 2009

nonn

a(n)/a(n-1) tends to (11+sqrt(21))/2 = 7.79128784...

For n>=2, a(n) equals 5^n times the permanent of the (2n-2)X(2n-2) tridiagonal matrix with 1/sqrt(5)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. [John M. Campbell, Jul 08 2011]

Indranil Ghosh, Table of n, a(n) for n = 0..1119
Index entries for linear recurrences with constant coefficients, signature (11,-25).

Cf. A165253.

LinearRecurrence[{11,-25},{1,5},30] (* Harvey P. Dale, Oct 02 2016 *)

G.f.: (1-6x)/(1-11x+25x^2).
a(n) = Sum_{k=0..n} A165253(n,k)*5^(n-k).
a(n) = ((21-sqrt(21))*(11+sqrt(21))^n+(21+sqrt(21))*(11-sqrt(21))^n )/(42*2^n). [Klaus Brockhaus, Sep 26 2009]

A165311 a(0)=1, a(1)=4, a(n)=9a(n-1)-16a(n-2) for n>1.

Original entry on oeis.org

1, 4, 20, 116, 724, 4660, 30356, 198644, 1302100, 8540596, 56031764, 367636340, 2412218836, 15827788084, 103854591380, 681446713076, 4471346955604, 29338975191220, 192509225431316, 1263159425822324, 8288287225499860

Offset: 0

Philippe Deléham, Sep 14 2009

nonn

a(n)/a(n-1) tends to (9+sqrt(17))/2 = 6.56155281...

Index entries for linear recurrences with constant coefficients, signature (9,-16).

LinearRecurrence[{9,-16},{1,4},30] (* Harvey P. Dale, Feb 19 2015 *)

G.f.: (1-5x)/(1-9x+16x^2). a(n)=Sum_{k, 0<=k<=n}A165253(n,k)*4^(n-k).

a(n) = ((17-sqrt(17))*(9+sqrt(17))^n+(17+sqrt(17))*(9-sqrt(17))^n )/(34*2^n). [From Klaus Brockhaus, Sep 26 2009]

A165314 a(0)=1, a(1)=6, a(n)=13a(n-1)-36a(n-2) for n>1.

Original entry on oeis.org

1, 6, 42, 330, 2778, 24234, 215034, 1923018, 17258010, 155125482, 1395342906, 12554940426, 112981880922, 1016786596650, 9150878043258, 82357097082954, 741210652521114, 6670882987788138, 60037895350485690, 540340851995941002

Offset: 0

Philippe Deléham, Sep 14 2009

nonn

a(n)/a(n-1) tends to 9.

G.f.: (1-7x)/(1-13x+36x^2). a(n)=Sum_{k, 0<=k<=n}A165253(n,k)*6^(n-k).

a(n) = (2*9^n+3*4^n)/5. [From Klaus Brockhaus, Sep 26 2009]

A165322 a(0)=1, a(1)=7, a(n)=15a(n-1)-49a(n-2) for n>1.

Original entry on oeis.org

1, 7, 56, 497, 4711, 46312, 463841, 4688327, 47596696, 484222417, 4931098151, 50239573832, 511969798081, 5217807853447, 53180597695736, 542036380617137, 5524696422165991, 56310663682250152, 573949830547618721

Offset: 0

Philippe Deléham, Sep 14 2009

nonn

a(n)/a(n-1) tends to (15+sqrt(29))/2=10,192582...

For n>=2, a(n) equals 7^n times the permanent of the (2n-2)X(2n-2) tridiagonal matrix with 1/sqrt(7)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. [John M. Campbell, Jul 08 2011]

Index entries for linear recurrences with constant coefficients, signature (15,-49).

Cf. A165253.

LinearRecurrence[{15,-49},{1,7},20] (* Harvey P. Dale, Jun 04 2021 *)

G.f.: (1-8x)/(1-15x+49x^2).
a(n) = Sum_{k=0..n} A165253(n,k)*7^(n-k).
a(n) = ((29-sqrt(29))*(15+sqrt(29))^n+(29+sqrt(29))*(15-sqrt(29))^n )/(58*2^n). [Klaus Brockhaus, Sep 26 2009]

A165323 a(0)=1, a(1)=8, a(n)=17a(n-1)-64a(n-2) for n>1.

Original entry on oeis.org

1, 8, 72, 712, 7496, 81864, 911944, 10263752, 116119368, 1317149128, 14959895624, 170020681416, 1932918264136, 21978286879688, 249924108049992, 2842099476549832, 32320548186147656, 367554952665320904

Offset: 0

Philippe Deléham, Sep 14 2009

a(n)/a(n-1) tends to (17+sqrt(33))/2 = 11.3722813...

For n>=2, a(n) equals 8^n times the permanent of the (2n-2) X (2n-2) tridiagonal matrix with 1/sqrt(8)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011

Index entries for linear recurrences with constant coefficients, signature (17,-64).

Cf. A165253.

LinearRecurrence[{17,-64},{1,8},20] (* Harvey P. Dale, Jun 08 2018 *)

G.f.: (1-9*x)/(1-17*x+64*x^2).
a(n) = Sum_{k=0..n} A165253(n,k)*8^(n-k).
a(n) = ((33-sqrt(33))*(17+sqrt(33))^n+(33+sqrt(33))*(17-sqrt(33))^n)/(66*2^n). - Klaus Brockhaus, Sep 28 2009

A165324 a(0)=1, a(1)=9, a(n)= 19a(n-1)-81a(n-2) for n>1.

Original entry on oeis.org

1, 9, 90, 981, 11349, 136170, 1667961, 20661489, 257463450, 3218224941, 40291734429, 504866733930, 6328837455921, 79353706214169, 995084584139610, 12478956895304901, 156498329695484709, 1962672755694512490

Offset: 0

Philippe Deléham, Sep 14 2009

nonn

a(n)/a(n-1) tends to (19+sqrt(37))/2 = 12.5413812...

Index entries for linear recurrences with constant coefficients, signature (19,-81).

LinearRecurrence[{19,-81},{1,9},20] (* Harvey P. Dale, Nov 20 2020 *)

G.f.: (1-10x)/(1-19x+81x^2). a(n)= Sum_{k, 0<=k<=n}A165253(n,k)*9^(n-k).

a(n) = ((37-sqrt(37))*(19+sqrt(37))^n+(37+sqrt(37))*(19-sqrt(37))^n)/(74*2^n). [From Klaus Brockhaus, Sep 28 2009]

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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A121314 Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

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A165310 a(0)=1, a(1)=3, a(n) = 7*a(n-1) - 9*a(n-2) for n > 1.

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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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A165312 a(0)=1, a(1)=5, a(n)=11*a(n-1)-25*a(n-2) for n>1.

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A165311 a(0)=1, a(1)=4, a(n)=9*a(n-1)-16*a(n-2) for n>1.

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A165314 a(0)=1, a(1)=6, a(n)=13*a(n-1)-36*a(n-2) for n>1.

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A165322 a(0)=1, a(1)=7, a(n)=15*a(n-1)-49*a(n-2) for n>1.

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A165310 a(0)=1, a(1)=3, a(n) = 7a(n-1) - 9a(n-2) for n > 1.

A165312 a(0)=1, a(1)=5, a(n)=11a(n-1)-25a(n-2) for n>1.

A165311 a(0)=1, a(1)=4, a(n)=9a(n-1)-16a(n-2) for n>1.

A165314 a(0)=1, a(1)=6, a(n)=13a(n-1)-36a(n-2) for n>1.

A165322 a(0)=1, a(1)=7, a(n)=15a(n-1)-49a(n-2) for n>1.

A165323 a(0)=1, a(1)=8, a(n)=17a(n-1)-64a(n-2) for n>1.

A165324 a(0)=1, a(1)=9, a(n)= 19a(n-1)-81a(n-2) for n>1.