cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A132610 Triangle T, read by rows, where row n+1 of T = row n of matrix power T^(2n) with appended '1' for n>=0 with T(0,0)=1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 14, 4, 1, 1, 194, 39, 6, 1, 1, 4114, 648, 76, 8, 1, 1, 118042, 15465, 1510, 125, 10, 1, 1, 4274612, 483240, 41121, 2908, 186, 12, 1, 1, 186932958, 18685905, 1424178, 89670, 4970, 259, 14, 1, 1, 9577713250, 861282832, 59857416, 3437248, 171700, 7824, 344, 16, 1, 1
Offset: 0

Views

Author

Paul D. Hanna, Aug 23 2007

Keywords

Examples

			Triangle begins:
1;
1, 1;
2, 1, 1;
14, 4, 1, 1;
194, 39, 6, 1, 1;
4114, 648, 76, 8, 1, 1;
118042, 15465, 1510, 125, 10, 1, 1;
4274612, 483240, 41121, 2908, 186, 12, 1, 1;
186932958, 18685905, 1424178, 89670, 4970, 259, 14, 1, 1; ...
GENERATE T FROM EVEN MATRIX POWERS OF T.
Matrix square T^2 begins:
1;
2, 1; <-- row 2 of T
5, 2, 1;
34, 9, 2, 1;
453, 88, 13, 2, 1; ...
where row 2 of T = row 1 of T^2 with appended '1'.
Matrix fourth powers T^4 begins:
1;
4, 1;
14, 4, 1; <-- row 3 of T
96, 22, 4, 1;
1215, 220, 30, 4, 1; ...
where row 3 of T = row 2 of T^4 with appended '1'.
Matrix sixth power T^6 begins:
1;
6, 1;
27, 6, 1;
194, 39, 6, 1; <-- row 4 of T
2394, 404, 51, 6, 1; ...
where row 4 of T = row 3 of T^6 with appended '1'.
ALTERNATE GENERATING METHOD.
Start with [1,0,0,0]; take partial sums and append 1 zero;
take partial sums twice more:
(1), 0, 0, 0;
1, 1, 1, (1), 0;
1, 2, 3, 4, (4);
1, 3, 6, 10, (14);
the final nonzero terms form row 3: [14,4,1,1].
Start with [1,0,0,0,0,0]; take partial sums and append 3 zeros;
take partial sums and append 1 zero; take partial sums twice more:
(1), 0, 0, 0, 0, 0;
1, 1, 1, 1, 1, (1), 0, 0, 0;
1, 2, 3, 4, 5, 6, 6, 6, (6), 0;
1, 3, 6, 10, 15, 21, 27, 33, 39, (39);
1, 4, 10, 20, 35, 56, 83, 116, 155, (194);
the final nonzero terms form row 4: [194,39,6,1,1].
Continuing in this way produces all the rows of this triangle.

Links

Crossrefs

Cf. columns: A132611, A132612, A132613; A132614; variants: A132615, A101479.
Cf. A304189, A304192, A304188.

Programs

  • Maple
    b:= proc(n) option remember;
      Matrix(n, (i,j)-> T(i-1,j-1))^(2*n-2)
    end:
T:= proc(n,k) option remember;
     `if`(n=k, 1, `if`(k>n, 0, b(n)[n,k+1]))
    end:
seq(seq(T(n,k), k=0..n), n=0..10);  # Alois P. Heinz, Apr 13 2020
  • Mathematica
    b[n_] := b[n] = MatrixPower[Table[T[i-1, j-1], {i, n}, {j, n}], 2n-2];
T[n_, k_] := T[n, k] = If[n == k, 1, If[k > n, 0, b[n][[n, k+1]]]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 27 2020, after Alois P. Heinz *)
  • PARI
    {T(n, k)=local(A=vector(n+1), p); A[1]=1; for(j=1, n-k-1, p=(n-1)^2-(n-j-1)^2; A=Vec((Polrev(A)+x*O(x^p))/(1-x))); A=Vec((Polrev(A)+x*O(x^p))/(1-x)); A[p+1]}
for(n=0,10, for(k=0,n, print1(T(n,k),", "));print(""))

Formula

T(n+1,1) is divisible by n for n>=1.

A304192 G.f. A(x) satisfies: [x^n] (1+x)^(n*(n+1)) / A(x) = 0 for n>0.

Original entry on oeis.org

1, 2, 7, 72, 1224, 29184, 892074, 33144288, 1445847756, 72291575784, 4070550314292, 254674699992768, 17518238545282080, 1313558965998605568, 106608039857256267192, 9309469431887521270848, 870250987085629018699728, 86703492688056304091302944, 9171254392641669833788501488, 1026466161170552167031522911104
Offset: 0

Views

Author

Paul D. Hanna, May 07 2018

Keywords

Comments

Note that: [x^n] (1+x)^(n*k) / G(x) = 0 for n>0 holds when G(x) = (1+x)/(1 - (k-1)*x) given some fixed k ; this sequence explores the case where k varies with n.

Examples

			G.f.: A(x) = 1 + 2*x + 7*x^2 + 72*x^3 + 1224*x^4 + 29184*x^5 + 892074*x^6 + 33144288*x^7 + 1445847756*x^8 + 72291575784*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in (1+x)^(n*(n+1)) / A(x) begins:
n=0: [1, -2, -3, -52, -955, -24246, -771113, -29428232, ...];
n=1: [1, 0, -6, -60, -1062, -26208, -820560, -30994704, ...];
n=2: [1, 4, 0, -80, -1337, -30840, -932010, -34438500, ...];
n=3: [1, 10, 39, 0, -1722, -39996, -1138680, -40521096, ...];
n=4: [1, 18, 147, 648, 0, -50832, -1503546, -50844384, ...];
n=5: [1, 28, 372, 3048, 15465, 0, -1898490, -67990260, ...];
n=6: [1, 40, 774, 9580, 83248, 483240, 0, -85539792, ...];
n=7: [1, 54, 1425, 24420, 303363, 2844270, 18685905, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1+x)^(n*(n+1)) / A(x) = 0 for n>0.
RELATED SEQUENCES.
The secondary diagonal in the above table that begins
[1, 4, 39, 648, 15465, 483240, 18685905, 861282832, 46085893011, ...]
yields A132612, column 1 of triangle A132610.
Related triangular matrix T = A132610 begins:
1;
1, 1;
2, 1, 1;
14, 4, 1, 1;
194, 39, 6, 1, 1;
4114, 648, 76, 8, 1, 1;
118042, 15465, 1510, 125, 10, 1, 1;
4274612, 483240, 41121, 2908, 186, 12, 1, 1;
186932958, 18685905, 1424178, 89670, 4970, 259, 14, 1, 1; ...
in which  row n+1 of T = row n of matrix power T^(2*n) with appended '1' for n>=0.

Links

Crossrefs

Cf. A132612, A304189, A304188, A304191, A304193, A132610.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); m=#A; A[m] = Vec( (1+x +x*O(x^m))^(m*(m-1))/Ser(A) )[m] ); A[n+1]}
for(n=0,30, print1(a(n),", "))

Formula

A132612(n+1) = [x^n] (1+x)^((n+1)*(n+2)) / A(x) for n>0.

A304184 G.f. A(x) satisfies: 0 = [x^n] (1+x)^(n*(n-1)/2) / A(x) for n>0.

Original entry on oeis.org

1, 0, 0, 1, 9, 117, 1851, 34923, 765933, 19155084, 538051164, 16771165230, 574424285076, 21443516818065, 866521903003641, 37683366660458208, 1754777541925339779, 87115221430910051901, 4592968693335470802627, 256294382115032521083411, 15090698035153332532531074
Offset: 0

Views

Author

Paul D. Hanna, May 08 2018

Keywords

Examples

			G.f.: A(x) = 1 + x^3 + 9*x^4 + 117*x^5 + 1851*x^6 + 34923*x^7 + 765933*x^8 + 19155084*x^9 + 538051164*x^10 + 16771165230*x^11 + 574424285076*x^12 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in (1+x)^(n*(n-1)/2) / A(x) begins:
n=0: [1, 0, 0, -1, -9, -117, -1850, -34905, -765618, ...];
n=1: [1, 0, 0, -1, -9, -117, -1850, -34905, -765618, ...];
n=2: [1, 1, 0, -1, -10, -126, -1967, -36755, -800523, ...];
n=3: [1, 3, 3, 0, -12, -147, -2229, -40815, -876000, ...];
n=4: [1, 6, 15, 19, 0, -180, -2706, -47955, -1005279, ...];
n=5: [1, 10, 45, 119, 191, 0, -3335, -59840, -1214055, ...];
n=6: [1, 15, 105, 454, 1341, 2646, 0, -73965, -1545531, ...];
n=7: [1, 21, 210, 1329, 5955, 19833, 46737, 0, -1913457, ...];
n=8: [1, 28, 378, 3275, 20438, 97533, 364936, 1003150, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] (1+x)^(n*(n-1)/2) / A(x) for n>0.
RELATED SEQUENCES.
The secondary diagonal in the above table that begins
[1, 1, 3, 19, 191, 2646, 46737, 1003150, 25330125,  ...]
yields A101481, column 0 of triangle A101479.
Related triangular matrix T = A101479 begins:
1;
1, 1;
1, 1, 1;
3, 2, 1, 1;
19, 9, 3, 1, 1;
191, 70, 18, 4, 1, 1;
2646, 795, 170, 30, 5, 1, 1;
46737, 11961, 2220, 335, 45, 6, 1, 1;
1003150, 224504, 37149, 4984, 581, 63, 7, 1, 1; ...
in which row n equals row (n-1) of T^(n-1) followed by '1' for n>0.

Crossrefs

Cf. A101481, A304185, A304186, A304187, A304189, A101479.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( (1+x +x*O(x^m))^((m-1)*(m-2)/2)/Ser(A) )[m] );A[n+1]}
for(n=0,30, print1(a(n),", "))

Formula

[x^n] (1+x)^(n*(n+1)/2) / A(x) = A101481(n+1) = A101479(n+1,0) for n>=0.
[x^n] (1+x)^((n+1)*(n+2)/2) / A(x) = Sum_{k=0..n} A101479(n+2,k+1) * A101479(k+1,0) for n>=0.

A304188 G.f. A(x) satisfies: [x^n] (1+x)^((n+1)*(n+2)) / A(x) = 0 for n>0.

Original entry on oeis.org

1, 6, 30, 264, 4179, 97758, 3000084, 113020056, 5018695542, 255724146876, 14671199172480, 934467807541824, 65366076594301044, 4978197982191048600, 409875168025688997456, 36268233577292228677728, 3431775207222740657912472, 345742547371677388835049744, 36948141363745699171977916032, 4174429749114285739841190548928
Offset: 0

Views

Author

Paul D. Hanna, May 09 2018

Keywords

Examples

			G.f.: A(x) = 1 + 6*x + 30*x^2 + 264*x^3 + 4179*x^4 + 97758*x^5 + 3000084*x^6 + 113020056*x^7 + 5018695542*x^8 + 255724146876*x^9 + 14671199172480*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in (1+x)^((n+1)*(n+2)) / A(x) begins:
n=0: [1, -4, -5, -114, -2289, -62568, -2113983, -84889290, ...];
n=1: [1, 0, -15, -154, -2790, -72432, -2378450, -93729900, ...];
n=2: [1, 6, 0, -224, -3924, -91776, -2858196, -109145280, ...];
n=3: [1, 14, 76, 0, -5310, -128964, -3714456, -134815824, ...];
n=4: [1, 24, 261, 1510, 0, -169752, -5223348, -178378752, ...];
n=5: [1, 36, 615, 6446, 41121, 0, -6779045, -251285430, ...];
n=6: [1, 50, 1210, 18696, 201435, 1424178, 0, -323428800, ...];
n=7: [1, 66, 2130, 44616, 675591, 7663626, 59857416, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1+x)^((n+1)*(n+2)) / A(x) = 0 for n>0.
RELATED SEQUENCES.
The secondary diagonal in the above table that begins
[1, 6, 76, 1510, 41121, 1424178, 59857416, 2957282370, ...]
yields A132613, column 2 of triangle A132610.
Related triangular matrix T = A132610 begins:
1;
1, 1;
2, 1, 1;
14, 4, 1, 1;
194, 39, 6, 1, 1;
4114, 648, 76, 8, 1, 1;
118042, 15465, 1510, 125, 10, 1, 1;
4274612, 483240, 41121, 2908, 186, 12, 1, 1;
186932958, 18685905, 1424178, 89670, 4970, 259, 14, 1, 1; ...
in which  row n+1 of T = row n of matrix power T^(2*n) with appended '1' for n>=0.

Crossrefs

Cf. A132613, A304189, A304192, A304193, A132610.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( (1+x +x*O(x^m))^(m*(m+1))/Ser(A) )[m] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

A132613(n+1) = [x^n] (1+x)^((n+2)*(n+3)) / A(x) for n>0.
Showing 1-4 of 4 results.

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