A193842 -id:A193842 - cp's OEIS frontend

A193860 Triangular array: the fission of ((2x+1)^n) by (q(n,x)), where q(n,x)=x^n+x^(n-1)+...+x+1.

1, 1, 5, 1, 7, 19, 1, 9, 33, 65, 1, 11, 51, 131, 211, 1, 13, 73, 233, 473, 665, 1, 15, 99, 379, 939, 1611, 2059, 1, 17, 129, 577, 1697, 3489, 5281, 6305, 1, 19, 163, 835, 2851, 6883, 12259, 16867, 19171, 1, 21, 201, 1161, 4521, 12585, 26025, 41385, 52905

Offset: 0

Clark Kimberling, Aug 07 2011

See A193842 for the definition of fission of two sequences of polynomials or triangular arrays.

			First six rows:
1
1...5
1...7....19
1...9....33...65
1...11...51...131...211
1...13...73...233...473...665

Cf. A193842, A193861. A193823.

z = 10;
p[n_, x_] := (2 x + 1)^n;
q[0, x_] := 1; q[n_, x_] := x*q[n - 1, x] + 1;
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193860 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]   (* A193861  *)

From Peter Bala, Jul 16 2013: (Start)
T(n,k) = sum {i = 0..k} binomial(n+1,k-i)*2^(k-i) for 0 <= k <= n.
O.g.f.: 1/( (1 - 3*x*t)*(1 - (2*x + 1)*t) ) = 1 + (1 + 5*x)*t + (1 + 7*x + 19*x^2)*t^2 + ....
The n-th row polynomial R(n,x) = 1/(1 - x)*( (2*x + 1)^(n+1) - (3*x)^(n+1) ). Cf. A193823. (End)

A094585 Triangle T of all positive differences of distinct Fibonacci numbers; also, triangle of all sums of consecutive distinct Fibonacci numbers.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 5, 8, 10, 11, 8, 13, 16, 18, 19, 13, 21, 26, 29, 31, 32, 21, 34, 42, 47, 50, 52, 53, 34, 55, 68, 76, 81, 84, 86, 87, 55, 89, 110, 123, 131, 136, 139, 141, 142, 89, 144, 178, 199, 212, 220, 225, 228, 230, 231, 144, 233, 288, 322, 343, 356, 364, 369, 372, 374, 375

Offset: 1

Clark Kimberling, May 13 2004

Row sums = (1,5,14,34,74,...) = A094584. Alternating row sums = (1,1,4,4,12,12,...) given by F(m+1)-1 if m is even and F(m+2)-1 if m is odd. Central numbers = (1,5,16,47,...) = A094586.

Let p(n,x) = Sum_{k=0..n} F(k+1)*x^(n-k) and q(n,x) = x * q(n-1,x)+1, with q(0,x)=1. Then A094585 is the fission of sequence (p(n,x)) by sequence (q(n,x)); see A193842 for the definition of fission. A094585 is the mirror of A193999. - Clark Kimberling, Aug 11 2011

			Rows 1 to 5:
  1;
  2,  3;
  3,  5,  6;
  5,  8, 10, 11;
  8, 13, 16, 18, 19;
T(5,4) = F(8) - F(4) = 21 - 3 = 18;
T(5,4) = F(6) + F(5) + F(4) + F(3) = 8 + 5 + 3 + 2 = 18.

Muniru A Asiru, Rows n=1..150 of triangle, flattened

Cf. A000045, A094584, A094586, A193999.

Flat(List([1..11],n->List([1..n],k->Fibonacci(n+3)-Fibonacci(n-k+3)))); # Muniru A Asiru, Apr 28 2019

(* See A193999. *)
Table[Fibonacci[n+3]-Fibonacci[n+3-k],{n,1,20}, {k,1,n}]//TableForm (* Rigoberto Florez, Oct 03 2019 *)

T(n, k) = F(n+3) - F(n+3-k) = F(n+1) + F(n) + ... + F(n+2-k), for k=1..n; n >= 1.

G.f.: x*y*(x*y+x+1)/((1-y*x)*(x^2+x-1)*(x^2*y^2+x*y-1)). - Vladimir Kruchinin, Jun 20 2025

A193846 Triangular array: the fission of ((x+2)^n) by ((x+1)^n).

Original entry on oeis.org

2, 4, 8, 8, 28, 26, 16, 80, 136, 80, 32, 208, 512, 568, 242, 64, 512, 1648, 2672, 2188, 728, 128, 1216, 4832, 10288, 12392, 8020, 2186, 256, 2816, 13312, 35072, 55648, 53216, 28432, 6560, 512, 6400, 35072, 110080, 216512, 273376, 216512, 98416

Offset: 0

Clark Kimberling, Aug 07 2011

See A193842 for the definition of fission of two sequences of polynomials or triangular arrays.

			First six rows:
2
4....8
8....28....26
16...80....136....80
32...208...512....568....242
64...512...1648...2672...2188...728

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Cf. A193842, A193847, A193848.

p[n_, x_] := (x + 2)^n;
q[n_, x_] := (x + 1)^n
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]   (* A193846 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A193847 *)
TableForm[Table[Reverse[h[n]/2], {n, 0, z}]]
Flatten[Table[Reverse[h[n]]/2, {n, -1, z}]] (* A193848 *)
TableForm[Table[h[n]/2, {n, 0, z}]]
Flatten[Table[h[n]/2, {n, -1, z}]]  (* A193849 *)

T(n)={[2*Vecrev(p) | p<-Vec(1/(1 - 2*(1 + 2*y)*x + y*(2 + 3*y)*x^2) + O(x*x^n))]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Feb 18 2024

G.f.: A(x,y) = 2/(1 - 2*(1 + 2*y)*x + y*(2 + 3*y)*x^2). - Andrew Howroyd, Feb 18 2024

A193850 Triangular array: the fission of ((x+2)^n) by (q(n,x)) given by q(n,x)=x^n+x^(n-1)+...+x+1.

Original entry on oeis.org

2, 4, 8, 8, 20, 26, 16, 48, 72, 80, 32, 112, 192, 232, 242, 64, 256, 496, 656, 716, 728, 128, 576, 1248, 1808, 2088, 2172, 2186, 256, 1280, 3072, 4864, 5984, 6432, 6544, 6560, 512, 2816, 7424, 12800, 16832, 18848, 19520, 19664, 19682, 1024, 6144

Offset: 0

Clark Kimberling, Aug 07 2011

See A193842 for the definition of fission of two sequences of polynomials or triangular arrays.

			First six rows:
2
4....8
8....20....26
16...48....72....80
32...112...192...232....242
64...256...496...656....716...728

Cf. A193842, A193850, A193852.

z = 10;
p[n_, x_] := (x + 2)^n;
q[0, x_] := 1; q[n_, x_] := x*q[n - 1, x] + 1;
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193850 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]   (* A193851  *)
TableForm[Table[Reverse[h[n]/2], {n, 0, z}]]
Flatten[Table[Reverse[h[n]]/2, {n, -1, z}]] (* A193852 *)
TableForm[Table[h[n]/2, {n, 0, z}]]
Flatten[Table[h[n]/2, {n, -1, z}]]  (* A193853 *)

A193858 Triangular array: the fission of ((x+1)^n) by ((2x+1)^n).

Original entry on oeis.org

1, 2, 3, 4, 10, 7, 8, 28, 34, 15, 16, 72, 124, 98, 31, 32, 176, 392, 444, 258, 63, 64, 416, 1136, 1672, 1404, 642, 127, 128, 960, 3104, 5616, 6152, 4092, 1538, 255, 256, 2176, 8128, 17440, 23536, 20488, 11260, 3586, 511, 512, 4864, 20608, 51136, 81952

Offset: 0

Clark Kimberling, Aug 07 2011

See A193842 for the definition of fission of two sequences of polynomials or triangular arrays.

			First six rows:
   1
   2     3
   4    10     7
   8    28    34    15
  16    72   124    98    31
  32   176   392   444   258   63

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Cf. A193842, A193859.

# The function 'fission' is defined in A193842.
A193858_row := n -> fission((n,x) -> (x+1)^n, (n,x) -> (2*x+1)^n, n);
for n from 0 to 5 do A193858_row(n) od; # Peter Luschny, Jul 23 2014

z = 10;
p[n_, x_] := (x + 1)^n;
q[n_, x_] := (2 x + 1)^n;
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* this sequence *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]   (* A193859 *)

T(n,k)={sum(j=0, k, 2^(n-j) * binomial(n-j, k-j))} \\ Andrew Howroyd, Feb 18 2024

# uses[fission from A193842]
A193858_row = lambda k: fission(lambda n,x: (x+1)^n, lambda n,x: (2*x+1)^n, k)
for n in range(7): A193858_row(n) # Peter Luschny, Jul 23 2014

From Andrew Howroyd, Feb 18 2024: (Start)
T(n,k) = Sum_{j=0..k} 2^(n-j) * binomial(n-j,k-j).
G.f.: A(x,y) = 1/(1 - (2 + 3*y)*x + 2*y*(1 + y)*x^2). (End)

A193856 Triangular array: the fission of (p(n,x)) by ((2x+1)^n), where p(n,x)=(x+1)^n.

Original entry on oeis.org

1, 1, 5, 1, 8, 19, 1, 11, 43, 65, 1, 14, 76, 194, 211, 1, 17, 118, 422, 793, 665, 1, 20, 169, 776, 2059, 3044, 2059, 1, 23, 229, 1283, 4387, 9221, 11191, 6305, 1, 26, 298, 1970, 8236, 22382, 38854, 39878, 19171, 1, 29, 376, 2864, 14146, 47090, 106000

Offset: 0

Clark Kimberling, Aug 07 2011

See A193842 for the definition of fission of two sequences of polynomials or triangular arrays.

			First six rows:
1
1...5
1...8....19
1...11...43....65
1...14...76....194...211
1...17...118...422...793...665

Cf. A193842, A193857, A193860.

T := (n,k) -> 3^k*binomial(n+1,k)*hypergeom([1,-k],[n-k+2], 1/3):
for n from 0 to 6 do seq(simplify(T(n,k)), k=0..n) od; # Peter Luschny, Nov 19 2018

z = 10;
p[n_, x_] := (2 x + 1)^n;
q[n_, x_] := (x + 1)^n;
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193856 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]   (* A193857 *)

From Peter Bala, Jul 16 2013: (Start)
T(n,k) = sum {i = 0..k} (-1)^k*binomial(n+1,k-i)*(-3)^(k-i) for 0 <= k <= n.
O.g.f.: 1/( (1 - 2*x*t)*(1 - (3*x + 1)*t) )= 1 + (1 + 5*x)*t + (1 + 8*x + 19*x^2)*t^2 + .... Cf. A193860.
The n-th row polynomial R(n,x) = 1/(x + 1)*( (3*x + 1)^(n+1) - (2*x)^(n+1) ). (End)
T(n, k) = 3^k*binomial(n+1, k)*hypergeom([1, -k], [n-k+2], 1/3). - Peter Luschny, Nov 19 2018

A193973 Triangular array: the fission of (p(n,x)) by (q(n,x)), where p(n,x)=xp(n-1,x)+n+1 with p(0,x)=1, and q(n,x)=xp(n-1,x)+1 with p(0,x)=1.

Original entry on oeis.org

2, 3, 5, 4, 7, 9, 5, 9, 12, 14, 6, 11, 15, 18, 20, 7, 13, 18, 22, 25, 27, 8, 15, 21, 26, 30, 33, 35, 9, 17, 24, 30, 35, 39, 42, 44, 10, 19, 27, 34, 40, 45, 49, 52, 54, 11, 21, 30, 38, 45, 51, 56, 60, 63, 65, 12, 23, 33, 42, 50, 57, 63, 68, 72, 75, 77, 13, 25, 36, 46

Offset: 0

Clark Kimberling, Aug 10 2011

See A193842 for the definition of fission of two sequences of polynomials or triangular arrays.

This array show the differences of the sequence of triangular numbers (A000217); viz., row n consists of t(n) - t(n-k) for k=1..n-1. - Clark Kimberling, Apr 15 2017

			First six rows:
  2;
  3,  5;
  4,  7,  9;
  5,  9, 12, 14;
  6, 11, 15, 18, 20;
  7, 13, 18, 22, 25, 27;
  ...

Clark Kimberling, Table of n, a(n) for n = 0..10000

Cf. A000217, A193664, A193842, A193974.

a000217 := proc(n) n*(n+1)/2 end:
seq(print(seq(a000217(n+2) - a000217(n+1-k),k=0..n)),n=0..5); # Georg Fischer, May 03 2022

z = 13;
p[0, x_] := 1; p[n_, x_] := x*p[n - 1, x] + n + 1;
q[0, x_] := 1; q[n_, x_] := x*q[n - 1, x] + 1;
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193973 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A193974 *)

T(n, k) = A000217(n + 2) - A000217(n + 1 - k), 0 <= k <= n. - Georg Fischer, May 03 2022

A193977 Triangular array: the fission of (p(n,x)) by (q(n,x)), where p(n,x)=xp(n-1,x)+n+1 with p(0,x)=1, and q(n,x)=sum{(k+1)x^k ; 0<=k<=n}.

Original entry on oeis.org

2, 6, 5, 12, 14, 9, 20, 27, 24, 14, 30, 44, 45, 36, 20, 42, 65, 72, 66, 50, 27, 56, 90, 105, 104, 90, 66, 35, 72, 119, 144, 150, 140, 117, 84, 44, 90, 152, 189, 204, 200, 180, 147, 104, 54, 110, 189, 240, 266, 270, 255, 224, 180, 126, 65, 132, 230, 297, 336

Offset: 0

Clark Kimberling, Aug 10 2011

See A193842 for the definition of fission of two sequences of polynomials or triangular arrays.

			First six rows:
2
6....5
12...14...9
20...27...24...14
30...44...45...36...20
42...65...72...66...50...27

Cf. A193842, A193978.

z = 11;
p[0, x_] := 1; p[n_, x_] := x*p[n - 1, x] + n + 1;
q[n_, x_] := Sum[(k + 1)*x^k, {k, 0, n}]
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193977 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A193978 *)

A194000 Triangular array: the self-fission of (p(n,x)), where sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).

Original entry on oeis.org

1, 2, 3, 3, 5, 9, 5, 8, 15, 24, 8, 13, 24, 39, 64, 13, 21, 39, 63, 104, 168, 21, 34, 63, 102, 168, 272, 441, 34, 55, 102, 165, 272, 440, 714, 1155, 55, 89, 165, 267, 440, 712, 1155, 1869, 3025, 89, 144, 267, 432, 712, 1152, 1869, 3024, 4895, 7920, 144, 233

Offset: 0

Clark Kimberling, Aug 11 2011

See A193917 for the self-fusion of the same sequence of polynomials.  (Fusion is defined at A193822; fission, at A193842; see A202503 and A202453 for infinite-matrix representations of fusion and fission.)
...
First five rows of P (triangle of coefficients of polynomials p(n,x)):
1
1...1
1...1...2
1...1...2...3
1...1...2...3...5
First eight rows of A194000:
1
2....3
3....5....9
5....8....15...24
8....13...24...39...64
13...21...29...63...104...168
21...34...63...102..168...272...441
34...55...102..165..272...440...714..1155
...
col 1:  A000045
col 2:  A000045
col 3:  A022086
col 4:  A022086
col 5:  A022091
col 6:  A022091
right edge, d(n,n):  A064831
d(n,n-1):  A059840
d(n,n-2):  A080097
d(n,n-3):  A080143
d(n,n-4):  A080144
...
Suppose n is an odd positive integer and d(n+1,x) is the polynomial matched to row n+1 of A194000 as in the Mathematica program (and definition of fission at A193842), where the first row is counted as row 0.

			First six rows:
1
2....3
3....5....9
5....8....15...24
8....13...24...39...64
13...21...29...63...104...168
...
Referring to the matrix product for fission at A193842,
the row (5,8,15,24) is the product of P(4) and QQ, where
P(4)=(p(4,4), p(4,3), p(4,2), p(4,1))=(5,3,2,1); and
QQ is the 4x4 matrix
(1..1..2..3)
(0..1..1..2)
(0..0..1..1)
(0..0..0..1).

Cf. A193842, A194001, A193917, A193918.

z = 11;
p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
q[n_, x_] := p[n, x];
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A194000 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A194001 *)

A193667 Triangular array: the fission of (p(n,x)) by (q(n,x)), where p(n,x)=(x+1)^n and q(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).

Original entry on oeis.org

1, 1, 3, 1, 4, 8, 1, 5, 12, 21, 1, 6, 17, 33, 55, 1, 7, 23, 50, 88, 144, 1, 8, 30, 73, 138, 232, 377, 1, 9, 38, 103, 211, 370, 609, 987, 1, 10, 47, 141, 314, 581, 979, 1596, 2584, 1, 11, 57, 188, 455, 895, 1560, 2575, 4180, 6765, 1, 12, 68, 245, 643, 1350, 2455

Offset: 0

Clark Kimberling, Aug 11 2011

See A193842 for the definition of the fission of P by Q, where P and Q are sequences of polynomials or triangular arrays (of coefficients of polynomials). A193667 is the mirror of A125172.

			First six rows:
1
1...3
1...4...8
1...5...12...21
1...6...17...33...55
1...7...23...50...88...144

Cf. A193842, A125172.

z = 11;
p[n_, x_] := (x + 1)^n;
q[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193667 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A125172 *)

cp's OEIS Frontend

A193860 Triangular array: the fission of ((2x+1)^n) by (q(n,x)), where q(n,x)=x^n+x^(n-1)+...+x+1.

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A094585 Triangle T of all positive differences of distinct Fibonacci numbers; also, triangle of all sums of consecutive distinct Fibonacci numbers.

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A193846 Triangular array: the fission of ((x+2)^n) by ((x+1)^n).

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A193850 Triangular array: the fission of ((x+2)^n) by (q(n,x)) given by q(n,x)=x^n+x^(n-1)+...+x+1.

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A193858 Triangular array: the fission of ((x+1)^n) by ((2x+1)^n).

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A193856 Triangular array: the fission of (p(n,x)) by ((2x+1)^n), where p(n,x)=(x+1)^n.

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A193973 Triangular array: the fission of (p(n,x)) by (q(n,x)), where p(n,x)=x*p(n-1,x)+n+1 with p(0,x)=1, and q(n,x)=x*p(n-1,x)+1 with p(0,x)=1.

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A193973 Triangular array: the fission of (p(n,x)) by (q(n,x)), where p(n,x)=xp(n-1,x)+n+1 with p(0,x)=1, and q(n,x)=xp(n-1,x)+1 with p(0,x)=1.

A193977 Triangular array: the fission of (p(n,x)) by (q(n,x)), where p(n,x)=xp(n-1,x)+n+1 with p(0,x)=1, and q(n,x)=sum{(k+1)x^k ; 0<=k<=n}.