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.

A136228 Triangle U, read by rows, where column k of U^(j+1) = column j of P^(3k+1) for j>=0, k>=0 and P=A136220.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 15, 24, 7, 1, 108, 198, 63, 10, 1, 1036, 2116, 714, 120, 13, 1, 12569, 28052, 9884, 1725, 195, 16, 1, 185704, 446560, 162729, 29190, 3393, 288, 19, 1, 3247546, 8325700, 3117660, 571225, 67756, 5880, 399, 22, 1, 65762269, 178284892
Offset: 0

Views

Author

Paul D. Hanna, Jan 28 2008

Keywords

Examples

			Triangle U begins:
1;
1, 1;
3, 4, 1;
15, 24, 7, 1;
108, 198, 63, 10, 1;
1036, 2116, 714, 120, 13, 1;
12569, 28052, 9884, 1725, 195, 16, 1;
185704, 446560, 162729, 29190, 3393, 288, 19, 1;
3247546, 8325700, 3117660, 571225, 67756, 5880, 399, 22, 1; ...
where column k of U = column 0 of P^(3k+1) and
triangle P = A136220 begins:
1;
1, 1;
3, 2, 1;
15, 10, 3, 1;
108, 75, 21, 4, 1;
1036, 753, 208, 36, 5, 1;
12569, 9534, 2637, 442, 55, 6, 1;
185704, 146353, 40731, 6742, 805, 78, 7, 1; ...
where column k of P = column 0 of U^(k+1).
Also, this triangle U can be obtained by the matrix product:
U = P * [P^2 shift right one column]
where P^2 shift right one column begins:
1;
0, 1;
0, 2, 1;
0, 8, 4, 1;
0, 49, 26, 6, 1;
0, 414, 232, 54, 8, 1;
0, 4529, 2657, 629, 92, 10, 1;
0, 61369, 37405, 9003, 1320, 140, 12, 1; ...

Crossrefs

Cf. A136221 (column 0), A136229 (column 1); related tables: A136220 (P), A136225 (P^2), A136230 (V), A136231 (W=P^3), A136217, A136218.

Programs

  • PARI
    {T(n,k)=local(P=Mat(1),U=Mat(1),PShR);if(n>0,for(i=0,n, PShR=matrix(#P,#P, r,c, if(r>=c,if(r==c,1,if(c==1,0,P[r-1,c-1]))));U=P*PShR^2; U=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,U[r,c], if(c==1,(P^3)[ #P,1],(P^(3*c-1))[r-c+1,1])))); P=matrix(#U, #U, r,c, if(r>=c, if(r<#R,P[r,c], (U^c)[r-c+1,1])))));U[n+1,k+1]}

Formula

This triangle U = P * [P^2 shift right one column] (see example), where P = A136220 and P^2 = A136225.

A136217 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms in row n at positions {floor(m*(m+7)/6), m>=0} and then taking partial sums, starting with all 1's in row 0.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 15, 8, 3, 1, 108, 49, 15, 4, 1, 1036, 414, 108, 24, 5, 1, 12569, 4529, 1036, 198, 34, 6, 1, 185704, 61369, 12569, 2116, 306, 46, 7, 1, 3247546, 996815, 185704, 28052, 3493, 453, 59, 8, 1, 65762269, 18931547, 3247546, 446560, 48800, 5555, 622, 74, 9, 1
Offset: 0

Views

Author

Paul D. Hanna, Dec 23 2007

Keywords

Comments

A variant of the triple factorial array A136212. Compare to triangle array A136218, which is generated by a complementary process.

Examples

			Square array begins:
(1),(1),1,(1),1,(1),1,(1),1,1,(1),1,1,(1),1,1,(1),1,1,(1),1,1,1,(1),...;
(1),(2),3,(4),5,(6),7,(8),9,10,(11),12,13,(14),15,16,(17),18,19,20,(21),..;
(3),(8),15,(24),34,(46),59,(74),90,108,(127),147,169,(192),216,242,(269),..;
(15),(49),108,(198),306,(453),622,(838),1080,1377,(1704),2062,2485,(2943),..;
(108),(414),1036,(2116),3493,(5555),8040,(11477),15483,20748,(26748),33528,..;
(1036),(4529),12569,(28052),48800,(82328),124335,(186261),260856,364551,..;
(12569),(61369),185704,(446560),811111,(1438447),2250731,(3513569),5078154,..;
(185704),(996815),3247546,(8325700),15684001,(29039188),46830722,...;
(3247546),(18931547),65762269,(178284892),346583419,...;
(65762269),(412345688),1515642725,(4317391240),...; ...
where terms in parenthesis are at positions {floor(m*(m+7)/6), m>=0} and are removed before taking partial sums to obtain the next row.
To generate the array, start with all 1's in row 0; from then on, obtain row n+1 from row n by first removing terms in row n at positions {floor(m*(m+7)/6), m>=0} and then taking partial sums.
For example, to generate row 2 from row 1:
[(1),(2),3,(4),5,(6),7,(8),9,10,(11),12,13,(14),15,16,(17),18,...],
remove terms at positions [0,1,3,5,7,10,13,16,20,...] to get:
[3, 5, 7, 9,10, 12,13, 15,16, 18,19,20, 22,23,24, 26,27,28,...]
then take partial sums to obtain row 2:
[3,8,15,24,34,46,59,74,90,108,127,147,169,192,216,242,269,...].
Continuing in this way will generate all the rows of this array.
Amazingly, column 0 of this array = column 0 of triangle P=A136220:
       1;
       1,      1;
       3,      2,     1;
      15,     10,     3,    1;
     108,     75,    21,    4,   1;
    1036,    753,   208,   36,   5,  1;
   12569,   9534,  2637,  442,  55,  6, 1;
  185704, 146353, 40731, 6742, 805, 78, 7, 1; ...
where column k of P^3 = column 0 of P^(3k+3) such that column 0 of P^3 = column 0 of P shift one place left.

Crossrefs

Cf. columns: A136221, A136226, A136229; related tables: A136220 (P), A136226 (P^2), A136232 (P^4).

Programs

  • Mathematica
    nmax = 9;
row[0] = Table[1, {nmax^2}];
row[n_] := row[n] = Accumulate[Delete[row[n-1], Table[{Floor[m((m+7)/6)+1] }, {m, 0, (1/2)(-7 + Sqrt[1 + 24 Length[row[n-1]]]) // Floor}]]];
R = row /@ Range[0, nmax];
T[n_, k_] := R[[n+1, k+1]];
Table[T[n-k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 06 2019 *)
  • PARI
    {T(n, k)=local(A=0, m=0, c=0, d=0); if(n==0, A=1, until(d>k, if(c==(m*(m+7))\6, m+=1, A+=T(n-1, c); d+=1); c+=1)); A}

Formula

Let triangular matrix P = A136220, then: column 0 (A136221) = column 0 of P; column 1 (A136226) = column 0 of P^2; column 3 (A136229) = column 0 of P^4.

A136230 Triangle V, read by rows, where column k of V^(j+1) = column j of P^(3k+2), for j>=0, k>=0 and where P=A136220.

Original entry on oeis.org

1, 2, 1, 8, 5, 1, 49, 35, 8, 1, 414, 325, 80, 11, 1, 4529, 3820, 988, 143, 14, 1, 61369, 54800, 14696, 2200, 224, 17, 1, 996815, 932761, 257264, 39468, 4123, 323, 20, 1, 18931547, 18426632, 5198680, 812801, 86506, 6919, 440, 23, 1, 412345688
Offset: 0

Views

Author

Paul D. Hanna, Jan 28 2008

Keywords

Examples

			This triangle V begins:
1;
2, 1;
8, 5, 1;
49, 35, 8, 1;
414, 325, 80, 11, 1;
4529, 3820, 988, 143, 14, 1;
61369, 54800, 14696, 2200, 224, 17, 1;
996815, 932761, 257264, 39468, 4123, 323, 20, 1;
18931547, 18426632, 5198680, 812801, 86506, 6919, 440, 23, 1; ...
where column k of V = column 0 of P^(3k+2) and
triangle P = A136220 begins:
1;
1, 1;
3, 2, 1;
15, 10, 3, 1;
108, 75, 21, 4, 1;
1036, 753, 208, 36, 5, 1;
12569, 9534, 2637, 442, 55, 6, 1; ...
where column k of P^2 = column 0 of V^(k+1).
Also, this triangle V equals the matrix product:
V = P^2 * [P shift right one column]
where P^2 = A136225 begins:
1;
2, 1;
8, 4, 1;
49, 26, 6, 1;
414, 232, 54, 8, 1;
4529, 2657, 629, 92, 10, 1;
61369, 37405, 9003, 1320, 140, 12, 1; ...
and P shift right one column begins:
1;
0, 1;
0, 1, 1;
0, 3, 2, 1;
0, 15, 10, 3, 1;
0, 108, 75, 21, 4, 1;
0, 1036, 753, 208, 36, 5, 1; ...
Also, this triangle V equals the matrix product:
V = U * [U shift down one row]
where triangle U = A136228 begins:
1;
1, 1;
3, 4, 1;
15, 24, 7, 1;
108, 198, 63, 10, 1;
1036, 2116, 714, 120, 13, 1; ...
and U shift down one row begins:
1;
1, 1;
1, 1, 1;
3, 4, 1, 1;
15, 24, 7, 1, 1;
108, 198, 63, 10, 1, 1;
1036, 2116, 714, 120, 13, 1, 1; ...

Crossrefs

Cf. A136226 (column 0), A136229 (column 1); related tables: A136220 (P), A136225 (P^2), A136230 (V), A136231 (W=P^3), A136234 (V^2), A136237 (V^3); A136217, A136218.

Programs

  • PARI
    {T(n,k)=local(P=Mat(1),U=Mat(1),V=Mat(1),PShR);if(n>0,for(i=0,n, PShR=matrix(#P,#P, r,c, if(r>=c,if(r==c,1,if(c==1,0,P[r-1,c-1])))); U=P*PShR^2;V=P^2*PShR; U=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,U[r,c], if(c==1,(P^3)[ #P,1],(P^(3*c-1))[r-c+1,1])))); V=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,V[r,c], if(c==1,(P^3)[ #P,1],(P^(3*c-2))[r-c+1,1])))); P=matrix(#U, #U, r,c, if(r>=c, if(r<#R,P[r,c], (U^c)[r-c+1,1])))));V[n+1,k+1]}

Formula

Triangle W=P^3=A136231 transforms column k of V into column k+1 of V. This triangle equals the matrix products: V = P^2 * [P shift right one column] and V = U * [U shift down one row] (see examples).

A136232 Triangle, read by rows, equal to the matrix 4th power of triangle A136220.

Original entry on oeis.org

1, 4, 1, 24, 8, 1, 198, 76, 12, 1, 2116, 888, 156, 16, 1, 28052, 12542, 2350, 264, 20, 1, 446560, 209506, 41034, 4864, 400, 24, 1, 8325700, 4058806, 821562, 100988, 8710, 564, 28, 1, 178284892, 89706276, 18631332, 2352116, 209440, 14168, 756, 32, 1
Offset: 0

Views

Author

Paul D. Hanna, Jan 28 2008

Keywords

Examples

			This triangle P^4 begins:
1,
4, 1;
24, 8, 1;
198, 76, 12, 1;
2116, 888, 156, 16, 1;
28052, 12542, 2350, 264, 20, 1;
446560, 209506, 41034, 4864, 400, 24, 1;
8325700, 4058806, 821562, 100988, 8710, 564, 28, 1;
178284892, 89706276, 18631332, 2352116, 209440, 14168, 756, 32, 1; ...
where column k = column 1 of U^(k+1);
triangle U = A136228 begins:
1;
1, 1;
3, 4, 1;
15, 24, 7, 1;
108, 198, 63, 10, 1;
1036, 2116, 714, 120, 13, 1;
12569, 28052, 9884, 1725, 195, 16, 1; ...
where column k of U = column 0 of P^(3k+1) and
triangle P = A136220 begins:
1;
1, 1;
3, 2, 1;
15, 10, 3, 1;
108, 75, 21, 4, 1;
1036, 753, 208, 36, 5, 1;
12569, 9534, 2637, 442, 55, 6, 1; ...

Crossrefs

Cf. A136229 (column 0); related tables: A136220 (P), A136228 (U).

Programs

  • PARI
    {T(n,k)=local(P=Mat(1),U,PShR);if(n>0,for(i=0,n, PShR=matrix(#P,#P, r,c, if(r>=c,if(r==c,1,if(c==1,0,P[r-1,c-1]))));U=P*PShR^2; U=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,U[r,c], if(c==1,(P^3)[ #P,1],(P^(3*c-1))[r-c+1,1])))); P=matrix(#U, #U, r,c, if(r>=c, if(r<#R,P[r,c], (U^c)[r-c+1,1])))));(P^4)[n+1,k+1]}

Formula

Column k of this triangle = column 1 of U^(k+1) where U = A136228.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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