A136220 -id:A136220 - cp's OEIS frontend

A136238 Matrix cube of triangle W = A136231; also equals P^9, where P = triangle A136220.

1, 9, 1, 99, 18, 1, 1323, 306, 27, 1, 21036, 5643, 621, 36, 1, 390012, 115917, 14580, 1044, 45, 1, 8287041, 2657946, 366129, 29754, 1575, 54, 1, 198918840, 67708113, 9968067, 882318, 52785, 2214, 63, 1, 5329794042, 1903562412, 294952140

Offset: 0

Paul D. Hanna, Feb 07 2008

			This triangle, W^3, begins:
1;
9, 1;
99, 18, 1;
1323, 306, 27, 1;
21036, 5643, 621, 36, 1;
390012, 115917, 14580, 1044, 45, 1;
8287041, 2657946, 366129, 29754, 1575, 54, 1;
198918840, 67708113, 9968067, 882318, 52785, 2214, 63, 1;
5329794042, 1903562412, 294952140, 27779046, 1804290, 85293, 2961, 72, 1;
where column 0 of W^3 = column 2 of W = triangle A136231.

Cf. related tables: A136220 (P), A136228 (U), A136230 (V), A136231 (W), A136235 (W^2).

{T(n,k)=local(P=Mat(1),U=Mat(1),W=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]))); W=P^3;));(W^3)[n+1,k+1]}

Column k of W^3 (this triangle) = column 2 of W^(k+1), where W = P^3 and P = triangle A136220.

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

Paul D. Hanna, Jan 28 2008

			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; ...

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

{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]}

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

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

Paul D. Hanna, Dec 23 2007

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

			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.

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

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 *)

{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}

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.

A136218 Triangle, read by rows of A136219(n) terms, where row n+1 is generated from row n by first inserting zeros in row n at positions {[m*(m+7)/6], m>=0} and then taking partial sums, starting with a '1' in row 0.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 3, 2, 2, 1, 1, 15, 15, 15, 12, 12, 9, 9, 6, 6, 4, 2, 2, 1, 108, 108, 108, 93, 93, 78, 78, 63, 63, 51, 39, 39, 30, 21, 21, 15, 9, 9, 5, 3, 1, 1, 1036, 1036, 1036, 928, 928, 820, 820, 712, 712, 619, 526, 526, 448, 370, 370, 307, 244, 244, 193, 154, 115, 115, 85

Offset: 0

Paul D. Hanna, Dec 23 2007

A variant of the triple factorial triangle A136213. Compare to square array A136217, which is generated by a complementary process.

			Triangle begins:
1;
1,1,1;
3,3,3,2,2,1,1;
15,15,15,12,12,9,9,6,6,4,2,2,1;
108,108,108,93,93,78,78,63,63,51,39,39,30,21,21,15,9,9,5,3,1,1;
1036,1036,1036,928,928,820,820,712,712,619,526,526,448,370,370,307,244,244,193,154,115,115,85,64,43,43,28,19,10,10,5,2,1;
12569,12569,12569,11533,11533,10497,10497,9461,9461,8533,7605,7605,6785,5965,5965,5253,4541,4541,3922,3396,2870,2870,2422,2052,1682,1682,1375,1131,887,887,694,540,425,310,310,225,161,118,75,75,47,28,18,8,8,3,1;
...
Number of terms in rows is given by A136219, which starts:
[1,3,7,13,22,33,47,64,84,106,131,159,190,224,261,301,343,388,...].
To generate row 3, start with row 2:
[3,3,3,2,2,1,1];
insert zeros at positions [0,1,3,5,7,10] to get:
[0,0,3,0,3,0,3,0,2,2,0,1,1],
then take reverse partial sums (from right to left) to obtain row 3:
[15,15,15,12,12,9,9,6,6,4,2,2,1].
For row 4, insert zeros in row 3 at positions [0,1,3,5,7,10,13,16,20]:
[0,0,15,0,15,0,15,0,12,12,0,9,9,0,6,6,0,4,2,2,0,1]
then take reverse partial sums to obtain row 4:
[108,108,108,93,93,78,78,63,63,51,39,39,30,21,21,15,9,9,5,3,1,1].
Continuing in this way will generate all the rows of this triangle.
Amazingly, column 0 of this triangle = 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.

Cf. A136221 (column 0), A136219; A136213, A136220.

{T(n,k)=local(A=[1],B);if(n>0,for(i=1,n,m=1;B=[0]; for(j=1,#A,if(j+m-1==(m*(m+7))\6,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])} /* for(n=0,6,for(k=0,2*n^2,if(T(n,k)==0,break,print1(T(n,k),",")));print("")) */

A136227 Column 1 of triangle A136225; also equals column 0 of triangle A136230.

Original entry on oeis.org

1, 4, 26, 232, 2657, 37405, 627435, 12248365, 273211787, 6862775083, 191840407156, 5909873159107, 199002812894375, 7273866200397039, 286882936292798852, 12145886485652450131, 549504341899436759416

Offset: 0

Paul D. Hanna, Jan 28 2008

nonn

Equals column 1 of P^2 (A136225) and equals column 0 of V^2, where P = A136220 and V = A136230 are triangular matrices such that column k of V = column 0 of P^(3k+2) and column j of P^2 = column 0 of V^(j+1).

Cf. A136226, A136225 (P^2), A136220 (P), A136230 (V); A136217.

/* Generate using matrix product recurrences of triangle P=A136220: */ {a(n)=local(P=Mat([1,0;1,1]),U,PShR);if(n>0,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]))));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^2)[n+2,2]}

A136233 Matrix square of triangle U = A136228, read by rows.

Original entry on oeis.org

1, 2, 1, 10, 8, 1, 75, 76, 14, 1, 753, 888, 196, 20, 1, 9534, 12542, 3087, 370, 26, 1, 146353, 209506, 55552, 7320, 598, 32, 1, 2647628, 4058806, 1136975, 159645, 14235, 880, 38, 1, 55251994, 89706276, 26224597, 3856065, 364403, 24480, 1216, 44, 1

Offset: 0

Paul D. Hanna, Feb 07 2008

			This triangle, U^2, begins:
1;
2, 1;
10, 8, 1;
75, 76, 14, 1;
753, 888, 196, 20, 1;
9534, 12542, 3087, 370, 26, 1;
146353, 209506, 55552, 7320, 598, 32, 1;
2647628, 4058806, 1136975, 159645, 14235, 880, 38, 1;
55251994, 89706276, 26224597, 3856065, 364403, 24480, 1216, 44, 1; ...
where column 0 of U^2 = column 1 of P = A136220.

Cf. A136222 (column 0); related tables: A136220 (P), A136228 (U), A136230 (V), A136231 (W=P^3), A136236 (U^3).

{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^2)[n+1,k+1]}

Column k of U^2 (this triangle) = column 1 of P^(3k+1), where P = triangle A136220.

A136234 Matrix square of triangle V = A136230, read by rows.

Original entry on oeis.org

1, 4, 1, 26, 10, 1, 232, 110, 16, 1, 2657, 1435, 248, 22, 1, 37405, 22135, 4240, 440, 28, 1, 627435, 397820, 81708, 9295, 686, 34, 1, 12248365, 8203057, 1773156, 214478, 17248, 986, 40, 1, 273211787, 191405232, 43039532, 5442349, 463267, 28747, 1340

Offset: 0

Paul D. Hanna, Feb 07 2008

			This triangle, V^2, begins:
1;
4, 1;
26, 10, 1;
232, 110, 16, 1;
2657, 1435, 248, 22, 1;
37405, 22135, 4240, 440, 28, 1;
627435, 397820, 81708, 9295, 686, 34, 1;
12248365, 8203057, 1773156, 214478, 17248, 986, 40, 1;
273211787, 191405232, 43039532, 5442349, 463267, 28747, 1340, 46, 1; ...
where column 0 of V^2 = column 1 of P^2 = triangle A136225.

Cf. A136227 (column 0); related tables: A136220 (P), A136228 (U), A136230 (V), A136231 (W=P^3), A136237 (V^3).

{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^2)[n+1,k+1]}

Column k of V^2 (this triangle) = column 1 of P^(3k+2), where P = triangle A136220.

A136236 Matrix cube of triangle U = A136228, read by rows.

Original entry on oeis.org

1, 3, 1, 21, 12, 1, 208, 156, 21, 1, 2637, 2350, 399, 30, 1, 40731, 41034, 8029, 750, 39, 1, 742620, 821562, 177198, 18865, 1209, 48, 1, 15624420, 18631332, 4317936, 502335, 36478, 1776, 57, 1, 372892266, 473187270, 115949841, 14390880, 1136811

Offset: 0

Paul D. Hanna, Feb 07 2008

			This triangle, U^3, begins:
1;
3, 1;
21, 12, 1;
208, 156, 21, 1;
2637, 2350, 399, 30, 1;
40731, 41034, 8029, 750, 39, 1;
742620, 821562, 177198, 18865, 1209, 48, 1;
15624420, 18631332, 4317936, 502335, 36478, 1776, 57, 1;
372892266, 473187270, 115949841, 14390880, 1136811, 62488, 2451, 66, 1;
where column 0 of U^3 = column 2 of P = A136220.

Cf. A136223 (column 0); related tables: A136220 (P), A136228 (U), A136230 (V), A136231 (W=P^3), A136233 (U^2).

{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^3)[n+1,k+1]}

Column k of U^3 (this triangle) = column 2 of P^(3k+1), where P = triangle A136220.

A136237 Matrix cube of triangle V = A136230, read by rows.

Original entry on oeis.org

1, 6, 1, 54, 15, 1, 629, 225, 24, 1, 9003, 3770, 504, 33, 1, 153276, 71655, 10988, 891, 42, 1, 3031553, 1539315, 259236, 23903, 1386, 51, 1, 68406990, 37072448, 6688092, 672672, 44135, 1989, 60, 1, 1736020806, 992226060, 188767184, 20225436, 1442049

Offset: 0

Paul D. Hanna, Feb 07 2008

			This triangle, V^3, begins:
1;
6, 1;
54, 15, 1;
629, 225, 24, 1;
9003, 3770, 504, 33, 1;
153276, 71655, 10988, 891, 42, 1;
3031553, 1539315, 259236, 23903, 1386, 51, 1;
68406990, 37072448, 6688092, 672672, 44135, 1989, 60, 1;
1736020806, 992226060, 188767184, 20225436, 1442049, 73304, 2700, 69, 1;
where column 0 of V^3 = column 2 of P^2 = triangle A136225.

Cf. related tables: A136220 (P), A136228 (U), A136230 (V), A136231 (W=P^3), A136234 (V^2).

{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^3)[n+1,k+1]}

Column k of V^3 (this triangle) = column 2 of P^(3k+2), where P = triangle A136220.

cp's OEIS Frontend

A136238 Matrix cube of triangle W = A136231; also equals P^9, where P = triangle A136220.

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

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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.

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A136218 Triangle, read by rows of A136219(n) terms, where row n+1 is generated from row n by first inserting zeros in row n at positions {[m*(m+7)/6], m>=0} and then taking partial sums, starting with a '1' in row 0.

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A136227 Column 1 of triangle A136225; also equals column 0 of triangle A136230.

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A136233 Matrix square of triangle U = A136228, read by rows.

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A136234 Matrix square of triangle V = A136230, read by rows.

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A136236 Matrix cube of triangle U = A136228, read by rows.

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A136237 Matrix cube of triangle V = A136230, read by rows.

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