A086645 -id:A086645 - cp's OEIS frontend

A165224 a(0)=1, a(1)=9, a(n) = 18a(n-1) - 49a(n-2) for n > 1.

1, 9, 113, 1593, 23137, 338409, 4957649, 72655641, 1064876737, 15607654857, 228758827313, 3352883803641, 49142725927201, 720277760311209, 10557006115168913, 154732499817791193, 2267891697076964737

Offset: 0

Philippe Deléham, Sep 08 2009

a(n)/a(n-1) tends to 9 + 4*sqrt(2) = 14.65685424... - Klaus Brockhaus, Sep 25 2009

Seiichi Manyama, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (18,-49).

Column k=8 of A333988.

Cf. A081294, A001541, A090965, A083884, A099140, A099141, A099142, A026244.

LinearRecurrence[{18,-49},{1,9},20] (* Harvey P. Dale, Sep 30 2016 *)

G.f.: (1-9x)/(1-18x+49x^2);
e.g.f.: exp(9x)*cosh(4*sqrt(2)x);
a(n) = Sum_{k=0..n} 8^k*binomial(2n,2k) = Sum_{k=0..n} 8^k*A086645(n,k);
a(n) = 7^n*T(n,9/7) where T is the Chebyshev polynomial of the first kind;
a(n) = (1+sqrt(8))^(2n)/2 + (1-sqrt(8))^(2n)/2.
a(n) = ((9-4*sqrt(2))^n + (9+4*sqrt(2))^n)/2. - Klaus Brockhaus, Sep 25 2009

A375440 Expansion of g.f. A(x) satisfying 0 = Sum_{k=0..n} (-1)^k * binomial(2n, 2k) * ([x^k] A(x)^n) for n >= 1.

Original entry on oeis.org

1, 1, 5, 65, 1593, 61953, 3476813, 265517441, 26492540401, 3349218907137, 523572565927509, 99215376614955457, 22415450137196941993, 5953820173628518544385, 1837040977427662958973341, 651657636773935012586716929, 263375512326578915885862469601, 120319850003020550647400856678401

Offset: 0

Paul D. Hanna, Sep 11 2024

nonn

Note that 0 = Sum_{k=0..n} (-1)^k * binomial(n+k, 2*k) * ([x^k] C(x)^n) for n >= 1 is satisfied by the Catalan function C(x) = 1 + x*C(x)^2 (A000108), where coefficient [x^k] C(x)^n = binomial(n+2*k-1,k)*n/(n+k).

			G.f.: A(x) = 1 + x + 5*x^2 + 65*x^3 + 1593*x^4 + 61953*x^5 + 3476813*x^6 + 265517441*x^7 + 26492540401*x^8 + ...
RELATED TABLES.
The table of coefficients of x^k in A(x)^n begins:
  n=1: [1, 1,  5,  65,  1593,  61953,  3476813, ...];
  n=2: [1, 2, 11, 140,  3341, 127742,  7097687, ...];
  n=3: [1, 3, 18, 226,  5259, 197637, 10869476, ...];
  n=4: [1, 4, 26, 324,  7363, 271928, 14799444, ...];
  n=5: [1, 5, 35, 435,  9670, 350926, 18895290, ...];
  n=6: [1, 6, 45, 560, 12198, 434964, 23165174, ...];
  ...
from which we may illustrate the defining property given by
0 = Sum_{k=0..n} (-1)^k * binomial(2*n, 2*k) * ([x^k] A(x)^n).
Using the coefficients in the table above, we see that
  n=1: 0 = 1*1 - 1*1;
  n=2: 0 = 1*1 - 6*2 + 1*11;
  n=3: 0 = 1*1 - 15*3 + 15*18 - 1*226;
  n=4: 0 = 1*1 - 28*4 + 70*26 - 28*324 + 1*7363;
  n=5: 0 = 1*1 - 45*5 + 210*35 - 210*435 + 45*9670 - 1*350926;
  n=6: 0 = 1*1 - 66*6 + 495*45 - 924*560 + 495*12198 - 66*434964 + 1*23165174;
  ...
The triangle A086645(n,k) = binomial(2*n, 2*k) begins:
  n=0: 1;
  n=1: 1,  1;
  n=2: 1,  6,   1;
  n=3: 1, 15,  15,   1;
  n=4: 1, 28,  70,  28,  1;
  n=5: 1, 45, 210, 210,  45,  1;
  n=6: 1, 66, 495, 924, 495, 66, 1;
  ...

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A086645, A375450, A375451.

{a(n) = my(A=[1],m); for(i=1, n, A=concat(A, 0); m=#A-1;
A[m+1] = sum(k=0, m, (-1)^(m-k+1) * binomial(2*m, 2*k) * polcoef(Ser(A)^m, k) )/m ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

a(n) ~ c * 2^(4*n) * n^(2*n + 1/2) / (Pi^(2*n) * exp(2*n)), where c = 7.23682343848882192289996... - Vaclav Kotesovec, Sep 12 2024

A178618 Triangle T(n,k) with the coefficient [x^k] of the series (1-x)^(n+1) * sum_{j=0..infinity} binomial(n+3j,3j)x^j, in row n, column k.

Original entry on oeis.org

1, 1, 2, 1, 7, 1, 1, 16, 10, 1, 30, 45, 5, 1, 50, 141, 50, 1, 1, 77, 357, 266, 28, 1, 112, 784, 1016, 266, 8, 1, 156, 1554, 3139, 1554, 156, 1, 1, 210, 2850, 8350, 6765, 1452, 55, 1, 275, 4917, 19855, 24068, 9042, 880, 11

Offset: 0

Roger L. Bagula, May 30 2010

Every third row is symmetrical.
Row sums are 3^n.
2*k instead of 3*k in the binomial() gives A034839 with alternating rows of A086645.

			1;
1, 2;
1, 7, 1;
1, 16, 10;
1, 30, 45, 5;
1, 50, 141, 50, 1;
1, 77, 357, 266, 28;
1, 112, 784, 1016, 266, 8;
1, 156, 1554, 3139, 1554, 156, 1;
1, 210, 2850, 8350, 6765, 1452, 55;
1, 275, 4917, 19855, 24068, 9042, 880, 11;

Cf. A094531, A111808, A027907.

A178618 := proc(n,k)
    (1-x)^(n+1)*add( binomial(n+3*j,3*j)*x^j,j=0..n+1) ;
    coeftayl(%,x=0,k) ;
end proc:
seq(seq(A178618(n,k),k=0..n),n=0..8) ; # R. J. Mathar, Nov 05 2012

p[x_, n_] = (-1)^(n + 1)*(-1 + x)^(n + 1)*Sum[Binomial[n + 3*k, 3*k]*x^k, {k, 0, Infinity}]
Flatten[Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]]

A280921 Degree of SO(n,C), the special orthogonal group, as an algebraic variety.

Original entry on oeis.org

2, 8, 40, 384, 4768, 111616, 3433600, 196968448, 14994641408, 2112561610752, 397713919469568, 137785594909556736, 64120367727755108352, 54666180849611078369280, 62864933930402036994048000, 131959858152100309567348408320, 374913851106401853810511580364800, 1938349609799484523235647407112847360, 13603397258157549964912652571654029312000

Offset: 2

Taylor Brysiewicz, Jan 10 2017

nonn

			For n = 4 we have a(4) = 2^3*det({6,1},{1,1}) = 2^3*(6-1) = 40.

M. Brandt, D. Bruce, T. Brysiewicz, R. Krone, E. Robeva, The degree of SO(n), arXiv:1701.03200 [math.AG], 2017

Cf. A086645, A103328, A280922, A280923.

a[n_] := 2^(n-1) Det[Table[Binomial[2n-2i-2j, n-2i], {i, n/2}, {j, n/2}]];
Table[a[n], {n, 2, 20}] (* Jean-François Alcover, Aug 12 2018 *)

a(n) = 2^(n-1)*matdet(matrix(n\2,n\2,i,j,binomial(2*n-2*i-2*j,n-2*i))); \\ Michel Marcus, Jan 14 2017

a(n) = 2^(n-1)*det(binomial(2n-2i-2j, n-2i))_{i,j=1..floor(n/2)}.
a(2*n+1) = A280922(n) * 2^(2*n).
Let M_n be the n X n matrix M_n(i, j) = binomial(2*i+2*j-2, 2*i-1) = A103328(i+j-1, i-1); then a(2*n+1) = 2^(2*n)*det(M_n).
Let M_n be the n X n matrix M_n(i,j) = binomial(2*i+2*j-4, 2*i-2) = A086645(i+j-2, i-1); then a(2*n) = 2^(2*n-1)*det(M_n).

A177809 Symmetrical sequence:Binomial(n,5*m).

Original entry on oeis.org

1, 1, 1, 1, 252, 1, 1, 3003, 3003, 1, 1, 15504, 184756, 15504, 1, 1, 53130, 3268760, 3268760, 53130, 1, 1, 142506, 30045015, 155117520, 30045015, 142506, 1, 1, 324632, 183579396, 3247943160, 3247943160, 183579396, 324632, 1, 1, 658008, 847660528, 40225345056

Offset: 0

Roger L. Bagula, Dec 13 2010

Row sums are A070782.

5th in the sequence of sequence Binomial(n,k*m),k=1,2,3,4,5,...

			{1},
{1, 1},
{1, 252, 1},
{1, 3003, 3003, 1},
{1, 15504, 184756, 15504, 1},
{1, 53130, 3268760, 3268760, 53130, 1},
{1, 142506, 30045015, 155117520, 30045015, 142506, 1},
{1, 324632, 183579396, 3247943160, 3247943160, 183579396, 324632, 1},
{1, 658008, 847660528, 40225345056, 137846528820, 40225345056, 847660528, 658008, 1},
{1, 1221759, 3190187286, 344867425584, 3169870830126, 3169870830126, 344867425584, 3190187286, 1221759, 1},
{1, 2118760, 10272278170, 2250829575120, 47129212243960, 126410606437752, 47129212243960, 2250829575120, 10272278170, 2118760, 1}

Cf. A086645, A034839, A070775, A177808, A139459, A070782.

t[n_, m_] = Binomial[n, 5*m];
Table[Table[t[n, m], {m, 0, Floor[n/5]}], {n, 0, 50, 5}];
Flatten[%]

A177810 Triangle binomial(6n,6m), 0 <= m <= n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 924, 1, 1, 18564, 18564, 1, 1, 134596, 2704156, 134596, 1, 1, 593775, 86493225, 86493225, 593775, 1, 1, 1947792, 1251677700, 9075135300, 1251677700, 1947792, 1, 1, 5245786, 11058116888, 353697121050, 353697121050, 11058116888, 5245786, 1

Offset: 0

Roger L. Bagula, Dec 13 2010

Row sums are A070967. k=6 in binomial(k*n,k*m) sequence similar to k=2 in A086645, k=4 in A070775,...

			1;
1, 1;
1, 924, 1;
1, 18564, 18564, 1;
1, 134596, 2704156, 134596, 1;
1, 593775, 86493225, 86493225, 593775, 1;

Cf. A034839, A177808, A139459, A070782, A177809, A070967.

t[n_, m_] := Binomial[n, 6*m]; Flatten@Table[Table[t[n, m], {m, 0, n/6}], {n, 0, 42, 6}]

Left-right symmetric: binomial(6*n,6*m) = binomial(6*n,6*(n-m)).

A232535 Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2n,2k) + binomial(2n+1,2k))/2.

Original entry on oeis.org

1, 1, 2, 1, 8, 3, 1, 18, 25, 4, 1, 32, 98, 56, 5, 1, 50, 270, 336, 105, 6, 1, 72, 605, 1320, 891, 176, 7, 1, 98, 1183, 4004, 4719, 2002, 273, 8, 1, 128, 2100, 10192, 18590, 13728, 4004, 400, 9, 1, 162, 3468, 22848, 59670, 68068, 34476, 7344, 561, 10, 1, 200, 5415

Offset: 0

Philippe Deléham, Nov 25 2013

Sum_{k=0..n}T(n,k)*x^k = A164111(n), A000012(n), A002001(n), A001653(n+1), A001019(n), A166965(n) for x =-1, 0, 1, 2, 4, 9 respectively.

			Triangle begins:
1
1, 2
1, 8, 3
1, 18, 25, 4
1, 32, 98, 56, 5
1, 50, 270, 336, 105, 6
1, 72, 605, 1320, 891, 176, 7
1, 98, 1183, 4004, 4719, 2002, 273, 8
1, 128, 2100, 10192, 18590, 13728, 4004, 400, 9

Cf. A086645, A091042
Cf. Columns : A000012, A001105, A180324 ; Diagonals: A000027, A131423
Cf. T(2*n,n): A228329, Row sums : A002001

T := (n,k) -> binomial(2*n, 2*k)*(2*n+1-k)/(2*n+1-2*k);
seq(seq(T(n,k), k=0..n), n=0..9); # Peter Luschny, Nov 26 2013

Flatten[Table[(Binomial[2n,2k]+Binomial[2n+1,2k])/2,{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jul 05 2015 *)

G.f.: (1-x)/(1-2*x*(1+y)+x^2*(1-y)^2).
T(n,k) = 2*T(n-1,k)+2*T(n-1,k-1)+2*T(n-2,k-1)-T(n-2,k)-T(n-2,k-2), T(0,0)=T(1,0)=1, T(1,1)=2, T(n,k)=0 if k<0 or if k>n.
T(n,k) = (A086645(n,k) + A091042(n,k))/2.
T(n,k) = binomial(2*n,2*k)*(2*n+1-k)/(2*n+1-2*k). - Peter Luschny, Nov 26 2013

A338523 Triangle T(n,m) = (2mn+2n-2m^2+1)C(2n+2,2m+1)/(4n+2).

Original entry on oeis.org

1, 2, 2, 3, 14, 3, 4, 44, 44, 4, 5, 100, 238, 100, 5, 6, 190, 828, 828, 190, 6, 7, 322, 2233, 4092, 2233, 322, 7, 8, 504, 5096, 14872, 14872, 5096, 504, 8, 9, 744, 10332, 43992, 70070, 43992, 10332, 744, 9, 10, 1050, 19176, 112200, 260780, 260780, 112200, 19176, 1050, 10

Offset: 0

Vladimir Kruchinin, Nov 01 2020

			1,
2, 2,
3, 14, 3,
4, 44, 44, 4,
5, 100, 238, 100, 5,
6, 190, 828, 828, 190, 6,
7, 322, 2233, 4092, 2233, 322, 7

Cf. A045623, A086645.

2nd column=2*A002412.

Table[Sum[Binomial[n + 1, 2 k + 1] Binomial[n - 2 k, m - k] (k + 1)*4^k, {k, 0, n} ], {n, 0, 9}, {m, 0, n}] // Flatten (* Michael De Vlieger, Nov 04 2020 *)

T(n,m):=((2*m*n+2*n-2*m^2+1)*binomial(2*n+2,2*m+1))/(4*n+2);

G.f.: (1/(1-x-x*y-4*x^2*y/(1-x-x*y)))^2.
T(n,m) = Sum_{k=0..n} C(n+1,2*k+1)*C(n-2*k,m-k)*(k+1)*4^k.
A045563(n) = (Sum_{m=0..n} T(n,m))/2^n.

cp's OEIS Frontend

A165224 a(0)=1, a(1)=9, a(n) = 18*a(n-1) - 49*a(n-2) for n > 1.

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A375440 Expansion of g.f. A(x) satisfying 0 = Sum_{k=0..n} (-1)^k * binomial(2*n, 2*k) * ([x^k] A(x)^n) for n >= 1.

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A178618 Triangle T(n,k) with the coefficient [x^k] of the series (1-x)^(n+1) * sum_{j=0..infinity} *binomial(n+3*j,3*j)*x^j, in row n, column k.

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A280921 Degree of SO(n,C), the special orthogonal group, as an algebraic variety.

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A177809 Symmetrical sequence:Binomial(n,5*m).

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A177810 Triangle binomial(6*n,6*m), 0 <= m <= n, read by rows.

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A232535 Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.

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A338523 Triangle T(n,m) = (2*m*n+2*n-2*m^2+1)*C(2*n+2,2*m+1)/(4*n+2).

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A165224 a(0)=1, a(1)=9, a(n) = 18a(n-1) - 49a(n-2) for n > 1.

A375440 Expansion of g.f. A(x) satisfying 0 = Sum_{k=0..n} (-1)^k * binomial(2n, 2k) * ([x^k] A(x)^n) for n >= 1.

A178618 Triangle T(n,k) with the coefficient [x^k] of the series (1-x)^(n+1) * sum_{j=0..infinity} binomial(n+3j,3j)x^j, in row n, column k.

A177810 Triangle binomial(6n,6m), 0 <= m <= n, read by rows.

A232535 Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2n,2k) + binomial(2n+1,2k))/2.

A338523 Triangle T(n,m) = (2mn+2n-2m^2+1)C(2n+2,2m+1)/(4n+2).