A135880 -id:A135880 - cp's OEIS frontend

A135899 Triangle, read by rows equal to the matrix product PR^-1P, where P = A135880 and R = A135894; PR^-1P equals triangle Q=A135885 shifted down one row.

1, 1, 1, 2, 1, 1, 6, 4, 1, 1, 25, 20, 6, 1, 1, 138, 126, 42, 8, 1, 1, 970, 980, 351, 72, 10, 1, 1, 8390, 9186, 3470, 748, 110, 12, 1, 1, 86796, 101492, 39968, 8936, 1365, 156, 14, 1, 1, 1049546, 1296934, 528306, 121532, 19090, 2250, 210, 16, 1, 1, 14563135, 18868652

Offset: 0

Paul D. Hanna, Dec 15 2007

			Triangle begins:
1;
1, 1;
2, 1, 1;
6, 4, 1, 1;
25, 20, 6, 1, 1;
138, 126, 42, 8, 1, 1;
970, 980, 351, 72, 10, 1, 1;
8390, 9186, 3470, 748, 110, 12, 1, 1;
86796, 101492, 39968, 8936, 1365, 156, 14, 1, 1;
1049546, 1296934, 528306, 121532, 19090, 2250, 210, 16, 1, 1; ...
This triangle equals matrix product P*R^-1*P,
which equals triangle Q shifted down one row,
where P = A135880 begins:
1;
1, 1;
2, 2, 1;
6, 7, 3, 1;
25, 34, 15, 4, 1;
138, 215, 99, 26, 5, 1;
970, 1698, 814, 216, 40, 6, 1; ...
and Q = P^2 = A135885 begins:
1;
2, 1;
6, 4, 1;
25, 20, 6, 1;
138, 126, 42, 8, 1;
970, 980, 351, 72, 10, 1;
8390, 9186, 3470, 748, 110, 12, 1; ...
and R = A135894 begins:
1;
1, 1;
2, 3, 1;
6, 12, 5, 1;
25, 63, 30, 7, 1;
138, 421, 220, 56, 9, 1;
970, 3472, 1945, 525, 90, 11, 1; ...
where column k of R equals column 0 of P^(2k+1),
and column k of Q=P^2 equals column 0 of P^(2k+2), for k>=0.

Cf. A135880 (P), A135885 (Q=P^2), A135894 (R); A135898 (P^-1*R), A135900 (R^-1*Q).

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

A135889 Column 0 of triangle A135888, which equals the matrix cube of triangle A135880; also equals column 1 of triangle A135894.

Original entry on oeis.org

1, 3, 12, 63, 421, 3472, 34380, 399463, 5344770, 81097517, 1377986373, 25947738574, 536726987593, 12104879913657, 295754724799758, 7784814503249896, 219682110287448760, 6617691928179590112

Offset: 0

Paul D. Hanna, Dec 15 2007

nonn

Cf. A135888, A135890; A135880, A135894.

{a(n)=local(P=Mat(1),R,PShR);if(n==0,1,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]))));R=P*PShR; R=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,R[r,c], if(c==1,(P^2)[ #P,1],(P^(2*c-1))[r-c+1,1])))); P=matrix(#R, #R, r,c, if(r>=c, if(r<#R,P[r,c], (R^c)[r-c+1,1]))));(P^3)[n+1,1])}

A135890 Column 1 of triangle A135888, which equals the matrix cube of triangle A135880; also equals column 1 of triangle A135895.

Original entry on oeis.org

1, 6, 39, 300, 2741, 29380, 363922, 5135894, 81557270, 1441771540, 28114817877, 600012111858, 13919315033624, 348932593149877, 9403371859278914, 271183690566871863, 8335374900994682248, 272083236017290793444

Offset: 0

Paul D. Hanna, Dec 15 2007

nonn

Cf. A135888, A135889; A135880, A135895.

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

A135891 Triangle, read by rows, equal to P^4, where triangle P = A135880; also equals Q^2, where triangle Q = P^2 = A135885.

Original entry on oeis.org

1, 4, 1, 20, 8, 1, 126, 64, 12, 1, 980, 580, 132, 16, 1, 9186, 6064, 1554, 224, 20, 1, 101492, 72832, 20260, 3240, 340, 24, 1, 1296934, 995050, 294218, 50496, 5830, 480, 28, 1, 18868652, 15301004, 4745522, 857840, 105620, 9516, 644, 32, 1, 308478492

Offset: 0

Paul D. Hanna, Dec 15 2007

			Triangle P^4 = Q^2 begins:
1;
4, 1;
20, 8, 1;
126, 64, 12, 1;
980, 580, 132, 16, 1;
9186, 6064, 1554, 224, 20, 1;
101492, 72832, 20260, 3240, 340, 24, 1;
1296934, 995050, 294218, 50496, 5830, 480, 28, 1;
18868652, 15301004, 4745522, 857840, 105620, 9516, 644, 32, 1;
308478492, 262203558, 84534154, 15907004, 2052450, 196400, 14490, 832, 36, 1;
where P = A135880 begins:
1;
1, 1;
2, 2, 1;
6, 7, 3, 1;
25, 34, 15, 4, 1;
138, 215, 99, 26, 5, 1;
970, 1698, 814, 216, 40, 6, 1; ...
and Q = P^2 = A135885 begins:
1;
2, 1;
6, 4, 1;
25, 20, 6, 1;
138, 126, 42, 8, 1;
970, 980, 351, 72, 10, 1;
8390, 9186, 3470, 748, 110, 12, 1; ...
where column k of Q equals column 0 of Q^(k+1) for k>=0
and column 0 of Q equals column 0 of P shift left.

Cf. A135886 (column 0); A135880 (P), A135885 (Q=P^2), A135893 (Q^3).

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

Column k of Q^2 = column 1 of Q^(k+1) for k>=0 where triangle Q = P^2 = A135885. Triangle P = A135880 is defined by: column k of P^2 equals column 0 of P^(2k+2) such that column 0 of P^2 equals column 0 of P shift left.

A136220 Triangle P, read by rows, where column k of P^3 equals column 0 of P^(3k+3) such that column 0 of P^3 equals column 0 of P shift one row up, with P(0,0)=1.

Original entry on oeis.org

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, 3247546, 2647628, 742620, 122350, 14330, 1325, 105, 8, 1, 65762269, 55251994, 15624420, 2571620

Offset: 0

Paul D. Hanna, Dec 25 2007, corrected Jan 24 2008

			Triangle P 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;
   3247546,  2647628,   742620,  122350,  14330,  1325,  105,   8, 1;
  65762269, 55251994, 15624420, 2571620, 298240, 26943, 2030, 136, 9, 1; ...
where column k of P = column 0 of U^(k+1) and U = A136228.
Matrix cube, W = P^3 (A136231), begins:
       1;
       3,     1;
      15,     6,     1;
     108,    48,     9,    1;
    1036,   495,    99,   12,   1;
   12569,  6338,  1323,  168,  15,  1;
  185704, 97681, 21036, 2754, 255, 18, 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 row up.
Matrix square, 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; ...
where column k of P^2 = column 0 of V^(k+1) and
triangle V = A136230 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; ...
where column k of V = column 0 of P^(3k+2).
Related 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 column k of P = column 0 of U^(k+1).
Surprisingly, column 0 of P is also found in square A136217:
(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),...;
(3),(8),15,(24),34,(46),59,(74),90,108,(127),147,169,(192),...;
(15),(49),108,(198),306,(453),622,(838),1080,1377,(1704),...;
(108),(414),1036,(2116),3493,(5555),8040,(11477),15483,...;
(1036),(4529),12569,(28052),48800,(82328),124335,(186261),...;
(12569),(61369),185704,(446560),811111,(1438447),2250731,...;
...
and has a recurrence similar to that of square array A136212
which generates the triple factorials.

Columns: A136221, A136222, A136223, A136224.
Related tables: A136228 (U), A136230 (V), A136231 (W=P^3), A136217, A136218.
Variants: A091351, A135880.

{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[n+1,k+1]}

Denote this triangle by P and define as follows.
Let [P^m]_k denote column k of matrix power P^m,
so that triangular matrix W = A136231 may be defined by
[W]_k = [P^(3k+3)]_0, for k>=0, such that
(1) W = P^3 and (2) [W]_0 = [P]_0 shift up one row.
Define the triangular matrix U = A136228 by
[U]_k = [P^(3k+1)]_0, for k>=0,
and the triangular matrix V = A136230 by
[V]_k = [P^(3k+2)]_0, for k>=0.
Then columns of P may be formed from powers of U:
[P]_k = [U^(k+1)]_0, for k>=0,
and columns of P^2 may be formed from powers of V:
[P^2]_k = [V^(k+1)]_0, for k>=0.
Further, columns of powers of P, U, V and W satisfy:
[U^(j+1)]_k = [P^(3k+1)]_j,
[V^(j+1)]_k = [P^(3k+2)]_j,
[W^(j+1)]_k = [P^(3k+3)]_j,
[W^(j+1)]_k = [W^(k+1)]_j,
[P^(3j+3)]_k = [P^(3k+3)]_j, for all j>=0, k>=0.
Also, we have the column transformations:
U * [P]k = [P]{k+1},
V * [P^2]k = [P^2]{k+1},
W * [P^3]k = [P^3]{k+1},
W * [U]k = [U]{k+1},
W * [V]k = [V]{k+1},
W * [W]k = [W]{k+1}, for all k>=0.
Other identities include the matrix products:
U = P * [P^2 shift right one column];
V = P^2 * [P shift right one column];
V = U * [U shift down one row];
W = V * [V shift down one row];
where the triangle transformations "shift right" and "shift down" are illustrated in examples of entries A136228 (U) and A136230 (V).

Typo in example corrected by Paul D. Hanna, Mar 27 2010

A135878 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms at positions [(m+3)^2/4 - 2] for m>=0 and then taking partial sums, starting with all 1's in row 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 6, 3, 1, 25, 25, 12, 4, 1, 138, 138, 63, 19, 5, 1, 970, 970, 421, 113, 28, 6, 1, 8390, 8390, 3472, 832, 190, 38, 7, 1, 86796, 86796, 34380, 7420, 1560, 283, 50, 8, 1, 1049546, 1049546, 399463, 78406, 15250, 2502, 411, 63, 9, 1, 14563135, 14563135

Offset: 0

Paul D. Hanna, Dec 14 2007

Column 0 is A135881 which equals column 0 of triangle A135879 and also equals column 0 of triangle A135880. Compare to triangle A135879, which is generated by a complementary process. An interesting variant is square array A135876, in which column 0 equals the double factorials (A001147).

			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,...;
(2),6,(12),19,(28),38,50,(63),77,93,(110),128,148,169,(191),214,...;
(6),25,(63),113,(190),283,411,(559),728,942,(1181),1446,1766,2116,...;
(25),138,(421),832,(1560),2502,3948,(5714),7830,10740,(14130),18036,...;
(138),970,(3472),7420,(15250),25990,44026,(67112),95918,138343,(189598),..;
(970),8390,(34380),78406,(174324),312667,(563287),897471,1329234,2003240,..;
(8390),86796,(399463),962750,(2291984),4295224,8168819,(13523882),20656067,.;
(86796),1049546,(5344770),13513589,(34169656),66534382,132787852,(227380975),.;
(1049546),14563135,(81097517),213885369,(570682050),1149537869,2395865161,..;
(14563135),228448504,(1377986373),3773851534,(10568874312),21945438536,...;
where terms in parenthesis are removed before taking partial sums.
For example, to generate row 2 from row 1, remove terms at positions
{[(m+3)^2/4-2], m>=0} = [0,2,4,7,10,14,18,23,28,34,...] to obtain:
[2, 4, 6,7, 9,10, 12,13,14, 16,17,18, 20,21,22,23, ...]
then take partial sums to get row 2:
[2, 6, 12,19, 28,38, 50,63,77, 93,110,128, 148,169,191,214, ...].
Repeating this process will generate all the rows of the triangle.
Triangle A135880 begins:
1;
1, 1;
2, 2, 1;
6, 7, 3, 1;
25, 34, 15, 4, 1;
138, 215, 99, 26, 5, 1;
970, 1698, 814, 216, 40, 6, 1; ...
and is generated by matrix powers of itself.

Cf. A135881, A135879, A135880; variants: A135876, A125714.

{T(n, k)=local(A=0, b=0, c=0, d=0); if(n==0, A=1, until(d>k, if(c==floor((b+3)^2/4)-2, b+=1, A+=T(n-1, c); d+=1); c+=1)); A}

A135879 Triangle, read by rows of A135901(n) terms, where row n+1 is generated from row n by inserting zeros at positions [(m+3)^2/4 - 2], as m=0,1,2,3,... and then taking partial sums from right to left, starting with a single 1 in row 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 6, 6, 4, 4, 2, 2, 1, 25, 25, 19, 19, 13, 13, 9, 5, 5, 3, 1, 1, 138, 138, 113, 113, 88, 88, 69, 50, 50, 37, 24, 24, 15, 10, 5, 5, 2, 1, 970, 970, 832, 832, 694, 694, 581, 468, 468, 380, 292, 292, 223, 173, 123, 123, 86, 62, 38, 38, 23, 13, 8, 3, 3, 1, 8390

Offset: 0

Paul D. Hanna, Dec 14 2007

Column 0 is A135881 which equals column 0 of square array A135878 and also equals column 0 of triangle A135880. Compare to square array A135878, which is generated by a complementary process. An interesting variant is triangle A135877 in which column 0 equals the double factorials (A001147).

			Triangle begins:
1;
1, 1;
2, 2, 1, 1;
6, 6, 4, 4, 2, 2, 1;
25, 25, 19, 19, 13, 13, 9, 5, 5, 3, 1, 1;
138, 138, 113, 113, 88, 88, 69, 50, 50, 37, 24, 24, 15, 10, 5, 5, 2, 1;
970, 970, 832, 832, 694, 694, 581, 468, 468, 380, 292, 292, 223, 173, 123, 123, 86, 62, 38, 38, 23, 13, 8, 3, 3, 1;
8390, 8390, 7420, 7420, 6450, 6450, 5618, 4786, 4786, 4092, 3398, 3398, 2817, 2349, 1881, 1881, 1501, 1209, 917, 917, 694, 521, 398, 275, 275, 189, 127, 89, 51, 51, 28, 15, 7, 4, 1, 1;
There are A135901(n) number of terms in row n.
To generate the triangle, start with a single 1 in row 0,
and then obtain row n+1 from row n by inserting zeros at
positions {[(m+3)^2/4 - 2], m=0,1,2,...} and then
taking reverse partial sums (i.e., summing from right to left).
Start with row 0, insert a zero in front of the '1' at position 0:
[0,1];
take reverse partial sums to get row 1:
[1,1];
insert zeros at positions [0,2]:
[0,1,0,1];
take reverse partial sums to get row 2:
[2,2,1,1];
insert zeros at positions [0,2,4]:
[0,2,0,2,0,1,1];
take reverse partial sums to get row 3:
[6,6,4,4,2,2,1];
insert zeros at positions [0,2,4,7]:
[0,6,0,6,0,4,4,0,2,2,0,1];
take reverse partial sums to get row 4:
[25,25,19,19,13,13,9,5,5,3,1,1];
insert zeros at positions [0,2,4,7,10,14]:
[0,25,0,25,0,19,19,0,13,13,0,9,5,5,0,3,1,1];
take reverse partial sums to get row 5:
[138,138,113,113,88,88,69,50,50,37,24,24,15,10,5,5,2,1].
Triangle A135880 begins:
1;
1, 1;
2, 2, 1;
6, 7, 3, 1;
25, 34, 15, 4, 1;
138, 215, 99, 26, 5, 1;
970, 1698, 814, 216, 40, 6, 1; ...
and is generated by matrix powers of itself.

Cf. A135881, A135901, A135878, A135880; variants: A135877, A127452, A125781.

{T(n,k)=local(A=[1],B);if(n>0,for(i=1,n,m=1;B=[]; for(j=1,#A,if(j+m-1==floor((m+2)^2/4)-1,m+=1;B=concat(B,0));B=concat(B,A[j])); A=Vec(Polrev(Vec(Pol(B)/(1-x+O(x^#B)))))));if(k+1>#A,0,A[k+1])}

A135900 Triangle, read by rows equal to the matrix product R^-1Q, where Q = A135885 and R = A135894; R^-1Q equals triangle R shifted down one row.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 6, 12, 5, 1, 1, 25, 63, 30, 7, 1, 1, 138, 421, 220, 56, 9, 1, 1, 970, 3472, 1945, 525, 90, 11, 1, 1, 8390, 34380, 20340, 5733, 1026, 132, 13, 1, 1, 86796, 399463, 247066, 72030, 13305, 1771, 182, 15, 1, 1, 1049546, 5344770, 3430936

Offset: 0

Paul D. Hanna, Dec 15 2007

			Triangle begins:
1;
1, 1;
1, 1, 1;
2, 3, 1, 1;
6, 12, 5, 1, 1;
25, 63, 30, 7, 1, 1;
138, 421, 220, 56, 9, 1, 1;
970, 3472, 1945, 525, 90, 11, 1, 1;
8390, 34380, 20340, 5733, 1026, 132, 13, 1, 1;
86796, 399463, 247066, 72030, 13305, 1771, 182, 15, 1, 1; ...
This triangle equals matrix product R^-1*Q,
which equals triangle R shifted down one row,
where P = A135880 begins:
1;
1, 1;
2, 2, 1;
6, 7, 3, 1;
25, 34, 15, 4, 1;
138, 215, 99, 26, 5, 1;
970, 1698, 814, 216, 40, 6, 1; ...
and Q = P^2 = A135885 begins:
1;
2, 1;
6, 4, 1;
25, 20, 6, 1;
138, 126, 42, 8, 1;
970, 980, 351, 72, 10, 1;
8390, 9186, 3470, 748, 110, 12, 1; ...
and R = A135894 begins:
1;
1, 1;
2, 3, 1;
6, 12, 5, 1;
25, 63, 30, 7, 1;
138, 421, 220, 56, 9, 1;
970, 3472, 1945, 525, 90, 11, 1; ...
where column k of R equals column 0 of P^(2k+1),
and column k of Q=P^2 equals column 0 of P^(2k+2), for k>=0.

Cf. A135880 (P), A135885 (Q=P^2), A135894 (R); A135898 (P^-1*R), A135899 (P*R^-1*P).

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

A135886 Column 1 of triangle Q = A135885; also equals column 0 of Q^2.

Original entry on oeis.org

1, 4, 20, 126, 980, 9186, 101492, 1296934, 18868652, 308478492, 5605768476, 112198139500, 2454071216496, 58267971181456, 1493114371576942, 41084194594171729, 1208473333806735096, 37849717704435895370

Offset: 0

Paul D. Hanna, Dec 15 2007

nonn

			Triangle Q = A135885 begins:
1;
2, 1;
6, 4, 1;
25, 20, 6, 1;
138, 126, 42, 8, 1;
970, 980, 351, 72, 10, 1;
8390, 9186, 3470, 748, 110, 12, 1; ...
where column k of Q equals column 0 of Q^(k+1) such that
column 0 of Q equals column 0 of P=A135880 shift left and Q=P^2.

Cf. A135885; other columns: A135881, A135887.

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

A135887 Column 2 of triangle Q = A135885; also equals column 0 of Q^3 = A135893.

Original entry on oeis.org

1, 6, 42, 351, 3470, 39968, 528306, 7906598, 132426050, 2457643895, 50110693656, 1114365815786, 26856942480503, 697612318151050, 19435260247394150, 578255661792065917, 18303904706366202568, 614296560055922433760

Offset: 0

Paul D. Hanna, Dec 15 2007

nonn

			Triangle Q = A135885 begins:
1;
2, 1;
6, 4, 1;
25, 20, 6, 1;
138, 126, 42, 8, 1;
970, 980, 351, 72, 10, 1;
8390, 9186, 3470, 748, 110, 12, 1; ...
where column k of Q equals column 0 of Q^(k+1) such that
column 0 of Q equals column 0 of P=A135880 shift left and Q=P^2.

Cf. A135885 (Q), A135893 (Q^3), A135880; other columns: A135881, A135886.

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

cp's OEIS Frontend

A135899 Triangle, read by rows equal to the matrix product P*R^-1*P, where P = A135880 and R = A135894; P*R^-1*P equals triangle Q=A135885 shifted down one row.

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PARI

A135889 Column 0 of triangle A135888, which equals the matrix cube of triangle A135880; also equals column 1 of triangle A135894.

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A135890 Column 1 of triangle A135888, which equals the matrix cube of triangle A135880; also equals column 1 of triangle A135895.

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A135891 Triangle, read by rows, equal to P^4, where triangle P = A135880; also equals Q^2, where triangle Q = P^2 = A135885.

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Formula

A136220 Triangle P, read by rows, where column k of P^3 equals column 0 of P^(3k+3) such that column 0 of P^3 equals column 0 of P shift one row up, with P(0,0)=1.

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Extensions

A135878 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms at positions [(m+3)^2/4 - 2] for m>=0 and then taking partial sums, starting with all 1's in row 0.

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A135879 Triangle, read by rows of A135901(n) terms, where row n+1 is generated from row n by inserting zeros at positions [(m+3)^2/4 - 2], as m=0,1,2,3,... and then taking partial sums from right to left, starting with a single 1 in row 0.

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A135900 Triangle, read by rows equal to the matrix product R^-1*Q, where Q = A135885 and R = A135894; R^-1*Q equals triangle R shifted down one row.

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PARI

A135886 Column 1 of triangle Q = A135885; also equals column 0 of Q^2.

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A135887 Column 2 of triangle Q = A135885; also equals column 0 of Q^3 = A135893.

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A135899 Triangle, read by rows equal to the matrix product PR^-1P, where P = A135880 and R = A135894; PR^-1P equals triangle Q=A135885 shifted down one row.

A135900 Triangle, read by rows equal to the matrix product R^-1Q, where Q = A135885 and R = A135894; R^-1Q equals triangle R shifted down one row.