A101822 -id:A101822 - cp's OEIS frontend

A084608 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+2x+3x^2)^n.

1, 1, 2, 3, 1, 4, 10, 12, 9, 1, 6, 21, 44, 63, 54, 27, 1, 8, 36, 104, 214, 312, 324, 216, 81, 1, 10, 55, 200, 530, 1052, 1590, 1800, 1485, 810, 243, 1, 12, 78, 340, 1095, 2712, 5284, 8136, 9855, 9180, 6318, 2916, 729, 1, 14, 105, 532, 2009, 5922, 13993, 26840, 41979

Offset: 0

Paul D. Hanna, Jun 01 2003

			Triangle begins:
  1;
  1,  2,  3;
  1,  4, 10,  12,   9;
  1,  6, 21,  44,  63,   54,   27;
  1,  8, 36, 104, 214,  312,  324,  216,   81;
  1, 10, 55, 200, 530, 1052, 1590, 1800, 1485, 810, 243;

Alois P. Heinz, Rows n = 0..100, flattened

Cf. A000079, A000244, A000400, A002426, A084600 - A084606, A006139, A084609 - A084615, A101822, A212697.

a084608 n = a084608_list !! n
a084608_list = concat $ iterate ([1,2,3] *) [1]
instance Num a => Num [a] where
   fromInteger k = [fromInteger k]
   (p:ps) + (q:qs) = p + q : ps + qs
   ps + qs         = ps ++ qs
   (p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
    *                = []
-- Reinhard Zumkeller, Apr 02 2011

A084608:= func< n,k | (&+[Binomial(n, k-j)*Binomial(k-j, j)*2^(k-2*j)*3^j: j in [0..k]]) >;
[A084608(n,k): k in [0..2*n], n in [0..13]]; // G. C. Greubel, Mar 27 2023

f:= proc(n) option remember; expand((1+2*x+3*x^2)^n) end:
T:= (n,k)-> coeff(f(n), x, k):
seq(seq(T(n, k), k=0..2*n), n=0..10);  # Alois P. Heinz, Apr 03 2011

row[n_] := (1+2x+3x^2)^n + O[x]^(2n+1) // CoefficientList[#, x]&; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 01 2017 *)

for(n=0,10, for(k=0,2*n,t=polcoeff((1+2*x+3*x^2)^n,k,x); print1(t",")); print(" "))

def A084608(n,k): return sum(binomial(n,j)*binomial(n-j,k-2*j)*2^(k-2*j)*3^j for j in range(k//2+1))
flatten([[A084608(n,k) for k in range(2*n+1)] for n in range(14)]) # G. C. Greubel, Mar 27 2023

From G. C. Greubel, Mar 27 2023: (Start)
T(n, k) = Sum_{j=0..k} binomial(n, k-j)*binomial(k-j, j)*2^(k-2*j)*3^j.
T(n, n) = A084609(n).
T(n, 2*n-1) = A212697(n), n >= 1.
T(n, 2*n) = A000244(n).
Sum_{j=0..2*n} T(n, k) = A000400(n).
Sum_{k=0..2*n} (-1)^k*T(n, k) = A000079(n).
Sum_{k=0..n} T(n-k, k) = A101822(n). (End)

A380886 Triangle T(n,k), 1<=k<=n: column k are the coefficients of the INVERT transform of Sum_{i=1..k} i*x^i.

Original entry on oeis.org

1, 1, 3, 1, 5, 8, 1, 11, 17, 21, 1, 21, 42, 50, 55, 1, 43, 100, 128, 138, 144, 1, 85, 235, 323, 358, 370, 377, 1, 171, 561, 813, 923, 965, 979, 987, 1, 341, 1331, 2043, 2378, 2510, 2559, 2575, 2584, 1, 683, 3158, 5150, 6125, 6527, 6681, 6737, 6755, 6765, 1, 1365, 7503, 12967, 15772, 16972, 17441, 17617, 17680, 17700, 17711

Offset: 1

R. J. Mathar, Feb 07 2025

			The full array starts
  1    1    1    1    1    1    1    1    1    1
  1    3    3    3    3    3    3    3    3    3
  1    5    8    8    8    8    8    8    8    8
  1   11   17   21   21   21   21   21   21   21
  1   21   42   50   55   55   55   55   55   55
  1   43  100  128  138  144  144  144  144  144
  1   85  235  323  358  370  377  377  377  377
  1  171  561  813  923  965  979  987  987  987
  1  341 1331 2043 2378 2510 2559 2575 2584 2584
  1  683 3158 5150 6125 6527 6681 6737 6755 6765
but the non-interesting upper right triangular part is not put into the sequence.

Cf. A001045 (column k=2), A101822 (column k=3), A322059 (column k=4?), A001906 (diagonal), A054452 (subdiagonal).

A380886 := proc(n,k)
    local g,x ;
    g := 1/(1-add(i*x^i,i=1..k)) ;
    coeftayl(g,x=0,n) ;
end proc:
seq(seq( A380886(n,k),k=1..n),n=1..12) ;

T(n,k) = [x^n] 1/(1-x^1-2*x^2-3*x^3-4*x^4-...-k*x^k) .

A100550 a(n) = a(n-1) + 2a(n-2) + 3a(n-3), for n>3, otherwise a(n) = n.

Original entry on oeis.org

0, 1, 2, 4, 11, 25, 59, 142, 335, 796, 1892, 4489, 10661, 25315, 60104, 142717, 338870, 804616, 1910507, 4536349, 10771211, 25575430, 60726899, 144191392, 342371480, 812934961, 1930252097, 4583236459, 10882545536, 25839774745, 61354575194

Offset: 0

gamo (gamo(AT)telecable.es), Nov 27 2004

A recursive and iterative algorithm for the computation of a(n) appear as Exercise 1.11 in the book Structure and Interpretation of Computer Programs. - Bas Kok (no(AT)spam.com), Jan 31 2008

Harold Abelson and Gerald Jay Sussman with Julie Sussman, Structure and Interpretation of Computer Programs, MIT Press, 1996.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,3).

Cf. A092836, A092835, A100477, A101822.

[n le 3 select n-1 else Self(n-1) +2*Self(n-2) +3*Self(n-3): n in [1..41]]; // G. C. Greubel, Mar 27 2023

LinearRecurrence[{1,2,3},{0,1,2},40] (* Harvey P. Dale, Mar 19 2023 *)

perl -e '@a=(0,1,2);for(3..30){$a[$]=$a[$-1]+2*$a[$-2]+3*$a[$-3];} print "@a ";'

@CachedFunction
def a(n): # a = A100550
    if (n<3): return n
    else: return a(n-1) + 2*a(n-2) + 3*a(n-3)
[a(n) for n in range(41)] # G. C. Greubel, Mar 27 2023

From R. J. Mathar, Aug 22 2008: (Start)
O.g.f.: x*(1+x)/(1-x-2*x^2-3*x^3).
a(n) = A101822(n-1) + A101822(n-2). (End)

A213947 Triangle read by rows: columns are finite differences of the INVERT transform of (1, 2, 3, ...) terms.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 1, 10, 6, 4, 1, 20, 21, 8, 5, 1, 42, 57, 28, 10, 6, 1, 84, 150, 88, 35, 12, 7, 1, 170, 390, 252, 110, 42, 14, 8, 1, 340, 990, 712, 335, 132, 49, 16, 9, 1, 682, 2475, 1992, 975, 402, 154, 56, 18, 10

Offset: 1

Gary W. Adamson, Jun 25 2012

Create an array in which the n-th row is the output of the INVERT transform on the first n natural numbers followed by zeros:
  1,   1,   1,   1,   1,   1,   1, ...
  1,   3,   5,  11,  21,  43,  85, ... (A001045)
  1,   3,   8,  17,  42, 100, 235, ... (A101822)
  1,   3,   8,  21,  50, 128, 323, ...
  ...
For example, row 3 is the INVERT transform of (1, 2, 3, 0, 0, 0, ...). Then, take finite differences of column terms starting from the top; which become the rows of the triangle.

			First few rows of the triangle:
  1;
  1,    2;
  1,    4,    3;
  1,   10,    6,    4;
  1,   20,   21,    8,    5;
  1,   42,   57,   28,   10,    6;
  1,   84,  150,   88,   35,   12,   7;
  1,  170,  390,  252,  110,   42,  14,   8;
  1,  340,  990,  712,  335,  132,  49,  16,  9;
  1,  682, 2475, 1992,  975,  402, 154,  56, 18, 10;
  1, 1364, 6138, 5464, 2805, 1200, 469, 176, 63, 20, 11;
  ...

Cf. A001906 (row sums), A026644 (2nd column).

read("transforms") ;
A213947i := proc(n,k)
        L := [seq(i,i=1..n),seq(0,i=0..k)] ;
        INVERT(L) ;
        op(k,%) ;
end proc:
A213947 := proc(n,k)
        if k = 1 then
                1;
        else
        A213947i(k,n)-A213947i(k-1,n) ;
        end if;
end proc: # R. J. Mathar, Jun 30 2012

A366942 Expansion of e.g.f. 1/(1-x-2x^2-3x^3).

Original entry on oeis.org

1, 1, 6, 48, 408, 5040, 72000, 1184400, 22619520, 482993280, 11459750400, 299495750400, 8531976499200, 263353163673600, 8754879893760000, 311808414677760000, 11845876873678848000, 478163414336864256000, 20436460099541950464000, 921972301728418676736000

Offset: 0

Enrique Navarrete, Oct 29 2023

nonn

a(n) is the number of ways to partition [n] into blocks of size at most 3, order the blocks, order the elements within each block, and choose 1 element from each block.
E.g.: a(4) = 408 since we have the following cases:
 1,2,3,4: 24 such orderings, 1 way to choose one element from each block;
 12,34: 24 such orderings, 2*2 ways to choose one element from each block;
 12,3,4: 72 such orderings, 2*1*1 ways to choose one element from each block;
 123,4: 48 such orderings, 3*1 ways to choose one element from each block;
so 24*1 + 24*4 + 72*2 + 48*3 = 408 ways.

Cf. A000073, A000142, A052567, A052585, A101822, A189886, A364324.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n, j)*j!*j, j=1..min(3, n)))
    end:
seq(a(n), n=0..19);  # Alois P. Heinz, Dec 14 2023

With[{m = 20}, Range[0, m]! * CoefficientList[Series[1/(1 - x - 2*x^2 - 3*x^3), {x, 0, m}], x]] (* Amiram Eldar, Oct 30 2023 *)

my(x='x+O('x^25)); Vec(serlaplace(1/(1-x-2*x^2-3*x^3))) \\ Michel Marcus, Oct 30 2023

a(n) = A000142(n)*A101822(n).

a(n) = n*(a(n-1)+(n-1)*(2*a(n-2)+(n-2)*3*a(n-3))) for n>=3. - Alois P. Heinz, Dec 14 2023

cp's OEIS Frontend

A084608 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+2*x+3*x^2)^n.

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A380886 Triangle T(n,k), 1<=k<=n: column k are the coefficients of the INVERT transform of Sum_{i=1..k} i*x^i.

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Maple

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A100550 a(n) = a(n-1) + 2*a(n-2) + 3*a(n-3), for n>3, otherwise a(n) = n.

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A213947 Triangle read by rows: columns are finite differences of the INVERT transform of (1, 2, 3, ...) terms.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

A366942 Expansion of e.g.f. 1/(1-x-2*x^2-3*x^3).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A084608 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+2x+3x^2)^n.

A100550 a(n) = a(n-1) + 2a(n-2) + 3a(n-3), for n>3, otherwise a(n) = n.

A366942 Expansion of e.g.f. 1/(1-x-2x^2-3x^3).