A133494 -id:A133494 - cp's OEIS frontend

A336142 Number of ways to choose a strict composition of each part of a strict integer partition of n.

1, 1, 1, 4, 6, 11, 22, 41, 72, 142, 260, 454, 769, 1416, 2472, 4465, 7708, 13314, 23630, 40406, 68196, 119646, 203237, 343242, 586508, 993764, 1677187, 2824072, 4753066, 7934268, 13355658, 22229194, 36945828, 61555136, 102019156, 168474033, 279181966

Offset: 0

Gus Wiseman, Jul 18 2020

nonn

A strict composition of n is a finite sequence of distinct positive integers summing to n.

			The a(1) = 1 through a(5) = 11 ways:
  (1)  (2)  (3)      (4)        (5)
            (1,2)    (1,3)      (1,4)
            (2,1)    (3,1)      (2,3)
            (2),(1)  (3),(1)    (3,2)
                     (1,2),(1)  (4,1)
                     (2,1),(1)  (3),(2)
                                (4),(1)
                                (1,2),(2)
                                (1,3),(1)
                                (2,1),(2)
                                (3,1),(1)

Alois P. Heinz, Table of n, a(n) for n = 0..7725

Multiset partitions of partitions are A001970.
Strict compositions are counted by A032020, A072574, and A072575.
Splittings of partitions are A323583.
Splittings of partitions with distinct sums are A336131.
Cf. A008289, A316245, A318684, A319794, A336128, A336130, A336132, A336135.
Partitions:
- Partitions of each part of a partition are A063834.
- Compositions of each part of a partition are A075900.
- Strict partitions of each part of a partition are A270995.
- Strict compositions of each part of a partition are A336141.
Strict partitions:
- Partitions of each part of a strict partition are A271619.
- Compositions of each part of a strict partition are A304961.
- Strict partitions of each part of a strict partition are A279785.
- Strict compositions of each part of a strict partition are A336142.
Compositions:
- Partitions of each part of a composition are A055887.
- Compositions of each part of a composition are A133494.
- Strict partitions of each part of a composition are A304969.
- Strict compositions of each part of a composition are A307068.
Strict compositions:
- Partitions of each part of a strict composition are A336342.
- Compositions of each part of a strict composition are A336127.
- Strict partitions of each part of a strict composition are A336343.
- Strict compositions of each part of a strict composition are A336139.

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 g(n$2):
seq(a(n), n=0..38);  # Alois P. Heinz, Jul 31 2020

strptn[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&];
Table[Length[Join@@Table[Tuples[Join@@Permutations/@strptn[#]&/@ctn],{ctn,strptn[n]}]],{n,0,20}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0,
     If[n == 0, p!, b[n, i-1, p] + b[n-i, Min[n-i, i-1], p+1]]];
g[n_, i_] := g[n, i] = If[i(i+1)/2 < n, 0,
     If[n == 0, 1, g[n, i-1] + b[i, i, 0]*g[n-i, Min[n-i, i-1]]]];
a[n_] := g[n, n];
a /@ Range[0, 38] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)

G.f.: Product_{k >= 1} (1 + A032020(k)*x^k).

A355382 Number of divisors d of n such that bigomega(d) = omega(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 02 2022

nonn

The statistic omega = A001221 counts distinct prime factors (without multiplicity).
The statistic bigomega = A001222 counts prime factors with multiplicity.
If positive integers are regarded as arrows from the number of prime factors to the number of distinct prime factors, this sequence counts divisible composable pairs. Is there a nice choice of a composition operation making this into an associative category?

			The set of divisors of 180 satisfying the condition is {12, 18, 20, 30, 45}, so a(180) = 5.

The version with multiplicity is A181591.
For partitions we have A355383, with multiplicity A339006.
The version for compositions is A355384.
Positions of first appearances are A355386.
A000005 counts divisors.
A001221 counts prime indices without multiplicity.
A001222 count prime indices with multiplicity.
A070175 gives representatives for bigomega and omega, triangle A303555.
Cf. A000712, A022811, A056239, A071625, A118914, A133494, A181819, A182850, A323014, A323022, A323023, A355385, A355388.

Table[Length[Select[Divisors[n],PrimeOmega[#]==PrimeNu[n]&]],{n,100}]

A140360 Inverse binomial transform of A140359.

Original entry on oeis.org

1, 0, 5, -5, 15, -25, 55, -105, 215, -425, 855, -1705, 3415, -6825, 13655, -27305, 54615, -109225, 218455, -436905, 873815, -1747625, 3495255, -6990505, 13981015, -27962025, 55924055, -111848105, 223696215, -447392425, 894784855, -1789569705, 3579139415

Offset: 0

Paul Curtz, Jun 24 2008

sign

For p*Jacobsthal numbers A001045, p=2: A078008 (A001045 differences, they are companions) or 1, 2*A001045(n), also in A133494; p=3: A062510; p=4: see A097073; p=6: A092297.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1, 2).

a:= n-> `if`(n=0, 1, (<<0|1>, <2|-1>>^(n-1). <<0,5>>)[1,1]):
seq(a(n), n=0..30);  # Alois P. Heinz, Dec 28 2010

{1}~Join~Table[(-5 (-1 + (-2)^(n - 1)))/3, {n, 32}] (* or *)
CoefficientList[Series[(-3 x^2 - x - 1)/(2 x^2 - x - 1), {x, 0, 32}], x] (* Michael De Vlieger, Apr 15 2016 *)

G.f.: (-3*x^2-x-1) / (2*x^2-x-1).
a(n) = (-5*(-1 + (-2)^(n-1)))/3, for n>0. - Andres Cicuttin, Apr 15 2016
a(n) = 5 - 2*a(n-1),  for n>2. - Andres Cicuttin, Apr 15 2016

More terms from Alois P. Heinz, Dec 28 2010

A193734 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(2x+1)^n and q(n,x)=(x+2)^n.

Original entry on oeis.org

1, 1, 2, 1, 6, 8, 1, 10, 32, 32, 1, 14, 72, 160, 128, 1, 18, 128, 448, 768, 512, 1, 22, 200, 960, 2560, 3584, 2048, 1, 26, 288, 1760, 6400, 13824, 16384, 8192, 1, 30, 392, 2912, 13440, 39424, 71680, 73728, 32768, 1, 34, 512, 4480, 25088, 93184, 229376, 360448, 327680, 131072

Offset: 0

Clark Kimberling, Aug 04 2011

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

Triangle T(n,k), read by rows, given by (1,0,0,0,0,0,0,0,...) DELTA (2,2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

			First six rows:
  1;
  1,  2;
  1,  6,   8;
  1, 10,  32,  32;
  1, 14,  72, 160, 128;
  1, 18, 128, 448, 768, 512;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A084938, A193722, A193735.

Cf. A005053, A026581, A081294, A133494.

function T(n, k) // T = A193734
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return k+1;
  else return T(n-1, k) + 4*T(n-1, k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 19 2023

(* First program *)
z = 8; a = 2; b = 1; c = 1; d = 2;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193734 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]      (* A193735 *)
(* Second program *)
T[n_, k_]:= T[n,k]= If[k<0 || k>n, 0, If[n<2, k+1, T[n-1,k] +4*T[n-1,k-1]]];
Table[T[n,k], {n,0,12}, {k,0,n}]//TableForm (* G. C. Greubel, Nov 19 2023 *)

def T(n, k): # T = A193734
    if (k<0 or k>n): return 0
    elif (n<2): return k+1
    else: return T(n-1, k) +4*T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 19 2023

T(n,k) = 4*T(n-1,k-1) + T(n-1,k) with T(0,0)=T(1,0)=1 and T(1,1)=2. - Philippe Deléham, Oct 05 2011
G.f.: (1 - 2*x*y)/(1 - x - 4*x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Nov 19 2023: (Start)
T(n, n) = A081294(n).
Sum_{k=0..n} T(n, k) = A005053(n).
Sum_{k=0..n} (-1)^k * T(n, k) = (-1)^n * A133494(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A026581(n-1) + (1/2)*[n=0]. (End)

A357186 Take the k-th composition in standard order for each part k of the n-th composition in standard order, then add up everything.

Original entry on oeis.org

0, 1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 4, 5, 5, 5, 4, 4, 5, 5, 4, 5, 5, 5, 3, 4, 5, 5, 4, 5, 5, 5, 5, 5, 6, 6, 5, 6, 6, 6, 4, 5, 5, 5, 5, 6, 6, 6, 5, 5, 6, 6, 5, 6, 6, 6, 3, 4, 5, 5, 5, 6, 6, 6, 5, 5, 6, 6, 5, 6, 6, 6, 5, 6, 6, 6, 6, 7, 7

Offset: 0

Gus Wiseman, Sep 28 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			Composition 92 in standard order is (2,1,1,3), with compositions ((2),(1),(1),(1,1)) so a(92) = 2 + 1 + 1 + 1 + 1 = 6.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
This is the sum of A029837 over the n-th composition in standard order.
Vertex degrees are A133494.
The version for Heinz numbers of partitions is A325033.
Row sums of A357135.
First differences are A357187.
Cf. A000120, A001511, A003963, A029931, A048896, A058891, A070939, A096111, A329395, A333766, A335404, A357137.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[stc/@stc[n]/.List->Plus,{n,0,100}]

a(n) = A029837(A357134(n)).

A357489 Numbers k such that the k-th composition in standard order is a triple (w,x,y) such that 2w = 3x + 4y.

Original entry on oeis.org

133, 1034, 4113, 8212, 32802, 65576, 131137, 262212, 524368, 1048706, 2097288, 4194464, 4194561, 8388868, 16777488, 33554752, 33554946, 67109384, 134218272, 134218753, 268436096, 268436484, 536871952, 1073742912, 1073743874, 2147484928, 2147485704, 4294969376

Offset: 1

Gus Wiseman, Nov 02 2022

nonn

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with the corresponding standard compositions begin:
     133: (5,2,1)
    1034: (7,2,2)
    4113: (8,4,1)
    8212: (9,2,3)
   32802: (10,4,2)
   65576: (11,2,4)
  131137: (11,6,1)
  262212: (12,4,3)
  524368: (13,2,5)

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
By sum, these triples appear to be counted by A008676.
The unordered version is A358102, counted by A357849.
A011782 counts compositions.
A066099 lists the standard compositions.
Cf. A000120, A029837, A029931, A070939, A133494.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,10000],Length[stc[#]]==3&&2*stc[#][[1]]==3*stc[#][[2]]+4*stc[#][[3]]&]

from itertools import count, islice
def A357489_gen(): # generator of terms
    for n in count(1):
        yield from sorted((1<0 and 6*(n-x)>m and m%6==0)
A357489_list = list(islice(A357489_gen(),40)) # Chai Wah Wu, Nov 02 2022

a(10)-a(28) from Chai Wah Wu, Nov 02 2022

A164948 Fibonacci matrix read by antidiagonals. (Inverse of A136158.)

Original entry on oeis.org

1, 1, -1, 3, -4, 1, 9, -15, 7, -1, 27, -54, 36, -10, 1, 81, -189, 162, -66, 13, -1, 243, -648, 675, -360, 105, -16, 1, 729, -2187, 2673, -1755, 675, -153, 19, -1, 2187, -7290, 10206, -7938, 3780, -1134, 210, -22, 1, 6561, -24057, 37908, -34020, 19278, -7182, 1764, -276, 25, -1, 19683, -78732, 137781, -139968, 91854, -40824, 12474, -2592, 351, -28, 1

Offset: 0

Mark Dols, Sep 01 2009

Triangle, read by rows, given by [1,2,0,0,0,0,0,0,0,...] DELTA [-1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 02 2009

			As triangle:
    1;
    1,   -1;
    3,   -4,    1;
    9,  -15,    7,   -1;
   27,  -54,   36,  -10,    1;
   81, -189,  162,  -66,   13,   -1;
  243, -648,  675, -360,  105,  -16,    1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000007, A000326, A001296, A001519, A002411, A003688, A006234.
Cf. A016777, A038763, A080420, A080421, A081294, A084938, A098399.
Cf. A133494, A136158, A164942.

A164948:= func< n,k | n eq 0 select 1 else (-1)^k*3^(n-k-1)*(n+2*k)*Binomial(n,k)/n >;
[A164948(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 26 2023

A164948[n_,k_]:= If[n==0,1,(-1)^k*3^(n-k-1)*(n+2*k)*Binomial[n,k]/n];
Table[A164948[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 26 2023 *)

def A164948(n,k): return 1 if (n==0) else (-1)^k*3^(n-k-1)*((n+2*k)/n)*binomial(n, k)
flatten([[A164948(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 26 2023

Sum_{k=0..n} T(n, k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A001519(n).
From Philippe Deléham, Oct 09 2011: (Start)
T(n,k) = 3*T(n-1,k) - T(n-1,k-1) with T(0,0)=1, T(1,0)=1, T(1,1)=-1.
Row n: Expansion of (1-x)*(3-x)^(n-1), n>0. (End)
G.f.: (1-2*x)/(1-3*x+x*y). - R. J. Mathar, Aug 12 2015
From G. C. Greubel, Dec 26 2023: (Start)
T(n, k) = (-1)^k * A136158(n, k).
T(n, k) = (-1)^k*3^(n-k-1)*((n+2*k)/n)*binomial(n, k), for n > 0, with T(0, 0) = 1.
T(n, 0) = A133494(n).
T(n, 1) = -A006234(n+2), n >= 1.
T(n, 2) = A080420(n-2), n >= 2.
T(n, 3) = -A080421(n-3), n >= 3.
T(2*n, n) = 4*(-1)^n*A098399(n-1) - (1/3)*[n=0].
T(n, n-4) = 27*(-1)^n*A001296(n-3), n >= 4.
T(n, n-3) = 9*(-1)^(n-1)*A002411(n-2), n >= 3.
T(n, n-2) = 3*(-1)^n*A000326(n-1) = (-1)^n*A062741(n-1), n >= 2.
T(n, n-1) = (-1)^(n-1)*A016777(n-1), n >= 1.
T(n, n) = (-1)^n.
Sum_{k=0..n} (-1)^k*T(n, k) = A081294(n).
Sum_{k=0..n} (-1)^k*T(n-k, k) = A003688(n). (End)

More terms from Philippe Deléham, Oct 09 2011

A193726 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(x+2)^n.

Original entry on oeis.org

1, 1, 2, 2, 9, 10, 4, 28, 65, 50, 8, 76, 270, 425, 250, 16, 192, 920, 2200, 2625, 1250, 32, 464, 2800, 9000, 16250, 15625, 6250, 64, 1088, 7920, 32000, 77500, 112500, 90625, 31250, 128, 2496, 21280, 103600, 315000, 612500, 743750, 515625, 156250

Offset: 0

Clark Kimberling, Aug 04 2011

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (2,3,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

			First six rows:
   1;
   1,   2;
   2,   9,  10;
   4,  28,  65,   50;
   8,  76, 270,  425,  250;
  16, 192, 920, 2200, 2625, 1250;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A002532, A011782, A020699, A045873, A081040, A084938.

Cf. A133494, A141413, A169634, A193722, A193727.

function T(n, k) // T = A193726
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return k+1;
  else return 2*T(n-1, k) + 5*T(n-1, k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 02 2023

(* First program *)
z = 8; a = 1; b = 2; c = 1; d = 2;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193726 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193727 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, k+1, 2*T[n-1, k] + 5*T[n -1, k-1]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 02 2023 *)

def T(n, k): # T = A193726
    if (k<0 or k>n): return 0
    elif (n<2): return k+1
    else: return 2*T(n-1, k) + 5*T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 02 2023

T(n,k) = 5*T(n-1,k-1) + 2*T(n-1,k) with T(0,0)=T(1,0)=1 and T(1,1)=2. - Philippe Deléham, Oct 05 2011
G.f.: (1-x-3*x*y)/(1-2*x-5*x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Dec 02 2023: (Start)
T(n, 0) = A011782(n).
T(n, n) = A020699(n).
T(n, n-1) = A081040(n-1).
Sum_{k=0..n} T(n, k) = A169634(n-1) + (4/7)*[n=0].
Sum_{k=0..n} (-1)^k * T(n, k) = (-1)^n*A133494(n) = -A141413(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A002532(n) + 2*A002532(n-1) + (3/5)*[n=0].
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A045873(n) - 2*A045873(n-1) + (3/5)*[n=0]. (End)

A193727 Mirror of the triangle A193726.

Original entry on oeis.org

1, 2, 1, 10, 9, 2, 50, 65, 28, 4, 250, 425, 270, 76, 8, 1250, 2625, 2200, 920, 192, 16, 6250, 15625, 16250, 9000, 2800, 464, 32, 31250, 90625, 112500, 77500, 32000, 7920, 1088, 64, 156250, 515625, 743750, 612500, 315000, 103600, 21280, 2496, 128

Offset: 0

Clark Kimberling, Aug 04 2011

This triangle is obtained by reversing the rows of the triangle A193726.

Triangle T(n,k), read by rows, given by (2,3,0,0,0,0,0,0,0,...) DELTA (1,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

			First six rows:
     1;
     2,    1;
    10,    9,    2;
    50,   65,   28,   4;
   250,  425,  270,  76,   8;
  1250, 2625, 2200, 920, 192; 16;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A011782, A015535, A020699, A081040, A084938.

Cf. A107839, A133494, A169634, A193722, A193726.

function T(n, k) // T = A193727
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return n-k+1;
  else return 5*T(n-1, k) + 2*T(n-1, k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 02 2023

(* First program *)
z = 8; a = 1; b = 2; c = 1; d = 2;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193726 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193727 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, n-k+1, 5*T[n-1, k] + 2*T[n-1, k-1]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 02 2023 *)

def T(n, k): # T = A193727
    if (k<0 or k>n): return 0
    elif (n<2): return n-k+1
    else: return 5*T(n-1, k) + 2*T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 02 2023

T(n,k) = A193726(n,n-k).
T(n,k) = 2*T(n-1,k-1) + 5*T(n-1,k) with T(0,0)=T(1,1)=1 and T(1,0)=2. - Philippe Deléham, Oct 05 2011
G.f.: (1-3*x-x*y)/(1-5*x-2*x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Dec 02 2023: (Start)
T(n, 0) = A020699(n).
T(n, 1) = A081040(n-1).
T(n, n) = A011782(n).
Sum_{k=0..n} T(n, k) = A169634(n-1) + (4/7)*[n=0].
Sum_{k=0..n} (-1)^k * T(n, k) = A133494(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = 2*A015535(n) + A015535(n-1) + (1/2)*[n=0].
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = 2*A107839(n-1) - A107839(n-2) + (1/2)*[n=0]. (End)

A193728 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (x+2)^n and q(n,x) = (2*x+1)^n.

Original entry on oeis.org

1, 2, 1, 8, 10, 3, 32, 64, 42, 9, 128, 352, 360, 162, 27, 512, 1792, 2496, 1728, 594, 81, 2048, 8704, 15360, 14400, 7560, 2106, 243, 8192, 40960, 87552, 103680, 73440, 31104, 7290, 729, 32768, 188416, 473088, 677376, 604800, 344736, 122472, 24786, 2187

Offset: 0

Clark Kimberling, Aug 04 2011

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

Triangle T(n,k), read by rows, given by (2,2,0,0,0,0,0,0,0,...) DELTA (1,2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

			First six rows:
    1;
    2,    1;
    8,   10,    3;
   32,   64,   42,    9;
  128,  352,  360,  162,  27;
  512, 1792, 2496, 1728, 594, 81;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000012, A000420, A060816, A081038, A081294.

Cf. A084938, A133494, A193722, A193729.

function T(n, k) // T = A193728
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return n-k+1;
  else return 4*T(n-1, k) + 3*T(n-1, k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 28 2023

(* First program *)
z = 8; a = 1; b = 2; c = 2; d = 1;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193728 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]   (* A193729 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, n-k+1, 4*T[n-1,k] + 3*T[n-1,k-1]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 28 2023 *)

def T(n, k): # T = A193728
    if (k<0 or k>n): return 0
    elif (n<2): return n-k+1
    else: return 4*T(n-1, k) + 3*T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 28 2023

T(n,k) = 3*T(n-1,k-1) + 4*T(n-1,k) with T(0,0)=T(1,1)=1 and T(1,0)=2. - Philippe Deléham, Oct 05 2011
G.f.: (1-2*x-2*x*y)/(1-4*x-3*x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Nov 28 2023: (Start)
T(n, n-k) = A193729(n, k).
T(n, 0) = A081294(n).
T(n, n-1) = 2*A081038(n-1).
T(n, n) = A133494(n).
Sum_{k=0..n} T(n, k) = (1/7)*(4*[n=0] + 3*A000420(n)).
Sum_{k=0..n} (-1)^k * T(n, k) = A000012(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = (5*b(n) + 4*b(n-1))/14 + (2/3)*[n=0].
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A060816(n),
where b(n) = (2 + sqrt(7))^n + (2 - sqrt(7))^n. (End)

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