A193722 -id:A193722 - cp's OEIS frontend

A193897 Triangular array: the self-fusion of (p(n,x)), where p(n,x)=sum{(k+1)*x^k : 0<=k<=n}.

1, 2, 1, 3, 6, 3, 4, 9, 12, 6, 5, 12, 18, 20, 10, 6, 15, 24, 30, 30, 15, 7, 18, 30, 40, 45, 42, 21, 8, 21, 36, 50, 60, 63, 56, 28, 9, 24, 42, 60, 75, 84, 84, 72, 36, 10, 27, 48, 70, 90, 105, 112, 108, 90, 45, 11, 30, 54, 80, 105, 126, 140, 144, 135, 110, 55, 12, 33

Offset: 0

Clark Kimberling, Aug 08 2011

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

			First six rows of A193897:
1
2...1
3...6....3
4...9....12...6
5...12...18...20...10
6...15...24...30...30...15

Cf. A193722, A193898.

z = 12;
p[n_, x_] := (n + 1)*x^n + p[n - 1, x] (* #7 *); p[0, x_] := 1;
q[n_, x_] := p[n, x];
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}]]  (* A193897 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193898 *)

A193917 Triangular array: the self-fusion of (p(n,x)), where p(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 3, 6, 9, 3, 5, 9, 15, 24, 5, 8, 15, 24, 40, 64, 8, 13, 24, 39, 64, 104, 168, 13, 21, 39, 63, 104, 168, 273, 441, 21, 34, 63, 102, 168, 272, 441, 714, 1155, 34, 55, 102, 165, 272, 440, 714, 1155, 1870, 3025, 55, 89, 165, 267, 440, 712, 1155

Offset: 0

Clark Kimberling, Aug 09 2011

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.  (Fusion is defined at A193822; fission, at A193742; 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 A193917:
1
1...1
1...2...3
2...3...6...9
3...5...9...15...24
5...8...15..24...40...64
8...13..24..39...64...104..168
13..21..39..63...104..168..273..441
...
col 1:  A000045
col 2:  A000045
col 3:  A022086
col 4:  A022086
col 5:  A022091
col 6:  A022091
col 7:  A022355
col 8:  A022355
right edge, w(n,n):  A064831
w(n,n-1):  A001654
w(n,n-2):  A064831
w(n,n-3):  A059840
w(n,n-4):  A080097
w(n,n-5):  A080143
w(n,n-6):  A080144
Suppose n is an even positive integer and w(n+1,x) is the polynomial matched to row n+1 of A193917 as in the Mathematica program (and definition of fusion at A193722), where the first row is counted as row 0.

			First six rows:
1
1...1
1...2...3
2...3...6....9
3...5...9....15...24
5...8...15...24...40...64

Cf. A193722, A064831, A193918, A194000, A194001.

z = 12;
p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
q[n_, x_] := p[n, x];
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}]]  (* A193917 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193918 *)

A193949 Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{F(k+1)x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers), and q(n,x)=sum{(k+1)(n+1)x^(n-k) : 0<=k<=n}.

Original entry on oeis.org

1, 2, 4, 3, 8, 13, 8, 19, 32, 45, 15, 38, 64, 92, 120, 30, 75, 128, 184, 242, 300, 56, 142, 243, 352, 464, 578, 692, 104, 264, 454, 659, 872, 1088, 1306, 1524, 189, 482, 831, 1210, 1604, 2006, 2411, 2818, 3225, 340, 869, 1502, 2191, 2910, 3644, 4386

Offset: 0

Clark Kimberling, Aug 10 2011

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

			First six rows:
1
2....4
3....8....13
8....19...32...45
15...38...64...92...120
30...75...128..184..242..300

Cf. A193722, A193950.

z = 12;
p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
q[n_, x_] := Sum[(k + 1) (n + 1)*x^(n - k), {k, 0, 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}]]  (* A193949 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193950 *)

A193955 Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{F(k+1)x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers), and q(n,x)=sum{((k+1)^2)x^(n-k) : 0<=k<=n}.

Original entry on oeis.org

1, 1, 4, 1, 5, 13, 2, 9, 23, 45, 3, 14, 36, 71, 120, 5, 23, 59, 116, 196, 300, 8, 37, 95, 187, 316, 484, 692, 13, 60, 154, 303, 512, 784, 1121, 1524, 21, 97, 249, 490, 828, 1268, 1813, 2465, 3225, 34, 157, 403, 793, 1340, 2052, 2934, 3989, 5219, 6625, 55

Offset: 0

Clark Kimberling, Aug 10 2011

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

			First six rows:
1
1...4
1...5....13
2...9....23...45
3...14...36...71....120
5...23...59...116...196...300

Cf. A193722, A193956.

z = 12;
p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
q[n_, x_] := Sum[((k + 1)^2)*x^(n - k), {k, 0, 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}]]  (* A193955 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193956 *)

A196665 Expansion of g.f. (1-6x)/(1-19x).

Original entry on oeis.org

1, 13, 247, 4693, 89167, 1694173, 32189287, 611596453, 11620332607, 220786319533, 4194940071127, 79703861351413, 1514373365676847, 28773093947860093, 546688785009341767, 10387086915177493573, 197354651388372377887, 3749738376379075179853

Offset: 0

Philippe Deléham, Oct 05 2011

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

Cf. A002001, A193722, A196661, A196662, A196663, A196664.

a(0) = 1, a(n) = 13*19^(n-1) for n>0.
a(n) = Sum_{k=0..n} A193722(n,k)*6^k.
From Elmo R. Oliveira, Mar 18 2025: (Start)
E.g.f.: (13*exp(19*x) + 6)/19.
a(n) = 19*a(n-1) for n > 1. (End)

A202674 Symmetric matrix based on (1,3,5,7,9,...), by antidiagonals.

Original entry on oeis.org

1, 3, 3, 5, 10, 5, 7, 18, 18, 7, 9, 26, 35, 26, 9, 11, 34, 53, 53, 34, 11, 13, 42, 71, 84, 71, 42, 13, 15, 50, 89, 116, 116, 89, 50, 15, 17, 58, 107, 148, 165, 148, 107, 58, 17, 19, 66, 125, 180, 215, 215, 180, 125, 66, 19, 21, 74, 143, 212, 265, 286, 265, 212

Offset: 1

Clark Kimberling, Dec 22 2011

Let s=(1,3,5,7,9,...) and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s.  Let T' be the transpose of T.  Then A202674 represents the matrix product M=T'*T.  M is the self-fusion matrix of s, as defined at A193722.  See A202675 for characteristic polynomials of principal submatrices of M.
...
row 1 (1,3,5,7,...) A005408
diagonal (1,10,35,84,...) A000447
antidiagonal sums (1,6,20,50,...) A002415

			Northwest corner:
1....3....5.....7.....9
3...10...18....26....34
5...18...35....53....71
7...26...53....84...116
9...34...71...116...165

Cf. A005408, A202675, A193722.

U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[2 k - 1, {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]

A202676 Symmetric matrix based on (1,4,7,10,13,...), by antidiagonals.

Original entry on oeis.org

1, 4, 4, 7, 17, 7, 10, 32, 32, 10, 13, 47, 66, 47, 13, 16, 62, 102, 102, 62, 16, 19, 77, 138, 166, 138, 77, 19, 22, 92, 174, 232, 232, 174, 92, 22, 25, 107, 210, 298, 335, 298, 210, 107, 25, 28, 122, 246, 364, 440, 440, 364, 246, 122, 28, 31, 137, 282, 430

Offset: 1

Clark Kimberling, Dec 22 2011

Let s=(1,4,7,10,13,...) and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s.  Let T' be the transpose of T.  Then A202676 represents the matrix product M=T'*T.  M is the self-fusion matrix of s, as defined at A193722.  See A202677 for characteristic polynomials of principal submatrices of M.
...
row 1 (1,4,7,10,...) A016777
diagonal (1,17,66,166,...) A024215
...
Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]: (1,25,144,484,..), the squares of the pentagonal numbers (A000326).

			Northwest corner:
1....4....7...10...13...16
4...17...32...47...62...77
7...32...66..102..138..174
10..47..102..166..232..298
13..62..138..232..335..440

Cf. A202677.

U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[3 k - 2, {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]

A202869 Symmetric matrix based on the lower Wythoff sequence, A000201, by antidiagonals.

Original entry on oeis.org

1, 3, 3, 4, 10, 4, 6, 15, 15, 6, 8, 22, 26, 22, 8, 9, 30, 39, 39, 30, 9, 11, 35, 54, 62, 54, 35, 11, 12, 42, 66, 87, 87, 66, 42, 12, 14, 47, 79, 108, 126, 108, 79, 47, 14, 16, 54, 90, 132, 159, 159, 132, 90, 54, 16, 17, 62, 103, 151, 196, 207, 196, 151, 103, 62

Offset: 1

Clark Kimberling, Dec 26 2011

Let s=(1,3,4,6,8,...)=A000201 and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A202869 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A202870 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
1...3....4....6....8....9
3...10...15...22...30...35
4...15...26...39...54...66
6...22...39...62...87...108
8...30...54...87...126..159

Cf. A202870.

s[k_] := Floor[k*GoldenRatio];
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}]   (* A054347 *)
Table[m[1, j], {j, 1, 12}]        (* A000201 *)

A202871 Symmetric matrix based on the Lucas sequence, A000032, by antidiagonals.

Original entry on oeis.org

1, 3, 3, 4, 10, 4, 7, 15, 15, 7, 11, 25, 26, 25, 11, 18, 40, 43, 43, 40, 18, 29, 65, 69, 75, 69, 65, 29, 47, 105, 112, 120, 120, 112, 105, 47, 76, 170, 181, 195, 196, 195, 181, 170, 76, 123, 275, 293, 315, 318, 318, 315, 293, 275, 123, 199, 445, 474, 510, 514

Offset: 1

Clark Kimberling, Dec 26 2011

Let s=(1,3,4,7,11,...)=A000201 and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A202871 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A202872 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
1....3....4....7....11...18
3....10...15...25...40...65
4....15...26...43...69...112
7....25...43...75...120..195
11...40...69...120..196..318

Cf. A202872.

s[k_] := LucasL[k];
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}] (* A027961 *)
Table[m[1, j], {j, 1, 12}]    (* A000032 *)

A202873 Symmetric matrix based on (1,3,7,15,31,...), by antidiagonals.

Original entry on oeis.org

1, 3, 3, 7, 10, 7, 15, 24, 24, 15, 31, 52, 59, 52, 31, 63, 108, 129, 129, 108, 63, 127, 220, 269, 284, 269, 220, 127, 255, 444, 549, 594, 594, 549, 444, 255, 511, 892, 1109, 1214, 1245, 1214, 1109, 892, 511, 1023, 1788, 2229, 2454, 2547, 2547, 2454

Offset: 1

Clark Kimberling, Dec 26 2011

Let s=(1,3,7,15,31,...) and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A202873 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A202767 for characteristic polynomials of principal submatrices of M.

			Northwest corner:
1.....3.....7...15...31.....63
3....10....24...52...108...220
7....24....59..129...269...549
15...52...129..284...594..1214
31...108..269..594..1245..2547

Cf. A202767.

s[k_] := -1 + 2^k;
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}] (* A000295, Eulerian *)
Table[m[1, j], {j, 1, 12}]    (* A000225 *)
Table[m[2, j], {j, 1, 12}]    (* A053208 *)

cp's OEIS Frontend

A193897 Triangular array: the self-fusion of (p(n,x)), where p(n,x)=sum{(k+1)*x^k : 0<=k<=n}.

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A193917 Triangular array: the self-fusion of (p(n,x)), where p(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).

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A193949 Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers), and q(n,x)=sum{(k+1)(n+1)*x^(n-k) : 0<=k<=n}.

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Mathematica

A193955 Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers), and q(n,x)=sum{((k+1)^2)*x^(n-k) : 0<=k<=n}.

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A196665 Expansion of g.f. (1-6*x)/(1-19*x).

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A202674 Symmetric matrix based on (1,3,5,7,9,...), by antidiagonals.

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A202676 Symmetric matrix based on (1,4,7,10,13,...), by antidiagonals.

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Mathematica

A202869 Symmetric matrix based on the lower Wythoff sequence, A000201, by antidiagonals.

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A202871 Symmetric matrix based on the Lucas sequence, A000032, by antidiagonals.

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Mathematica

A193949 Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{F(k+1)x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers), and q(n,x)=sum{(k+1)(n+1)x^(n-k) : 0<=k<=n}.

A193955 Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{F(k+1)x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers), and q(n,x)=sum{((k+1)^2)x^(n-k) : 0<=k<=n}.

A196665 Expansion of g.f. (1-6x)/(1-19x).