A114172
Triangle, read by rows, where the g.f. of column n, C_n(x), equals the g.f. of row n, R_n(x), divided by (1-x)^(n+1), for n>=0; e.g., C_n(x) = R_n(x)/(1-x)^(n+1).
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 1, 5, 6, 1, 1, 7, 16, 9, 1, 1, 9, 31, 36, 12, 1, 1, 11, 51, 95, 66, 15, 1, 1, 13, 76, 199, 229, 106, 18, 1, 1, 15, 106, 361, 601, 467, 156, 21, 1, 1, 17, 141, 594, 1316, 1509, 844, 216, 24, 1, 1, 19, 181, 911, 2542, 3951, 3293, 1395, 286, 27, 1, 1, 21, 226
Offset: 0
Triangle begins:
1;
1,1;
1,3,1;
1,5,6,1;
1,7,16,9,1;
1,9,31,36,12,1;
1,11,51,95,66,15,1;
1,13,76,199,229,106,18,1;
1,15,106,361,601,467,156,21,1;
1,17,141,594,1316,1509,844,216,24,1;
1,19,181,911,2542,3951,3293,1395,286,27,1;
1,21,226,1325,4481,8910,10193,6447,2155,366,30,1; ...
Where g.f. for columns is formed from g.f. of rows:
GF(column 2) = (1 + 3*x + 1*x^2)/(1-x)^3
= 1 + 6*x + 16*x^2 + 31*x^3 + 51*x^4 + 76*x^5 +...
GF(column 3) = (1 + 5*x + 6*x^2 + 1*x^3)/(1-x)^4
= 1 + 9*x + 36*x^2 + 95*x^3 + 199*x^4 + 361*x^5 +...
GF(column 4) = (1 + 7*x + 16*x^2 + 9*x^3 + 1*x^4)/(1-x)^5
= 1 + 12*x + 66*x^2 + 229*x^3 + 601*x^4 + 1316*x^5 +...
A114175
Sums of squared terms in rows of triangle A114172.
Original entry on oeis.org
1, 2, 11, 63, 388, 2484, 16330, 109549, 745851, 5141513, 35807609, 251514544, 1779289072, 12663046683, 90583092779, 650818310772, 4693790254019, 33965214382094, 246504409256254, 1793728983702287, 13083256171763427
Offset: 0
A114173
Row sums of triangle A114172, where the g.f. of column n equals the g.f. of row n divided by (1-x)^(n+1).
Original entry on oeis.org
1, 2, 5, 13, 34, 90, 240, 643, 1729, 4663, 12607, 34156, 92702, 251975, 685759, 1868310, 5094753, 13904028, 37971570, 103762963, 283703065, 776069296, 2123892683, 5814922585, 15926507984, 43636524179, 119597410199
Offset: 0
A114176
Triangle, read by rows, where the g.f. of column n, C_n(x), equals the g.f. of row n, R_n(x), divided by (1-x)^(n+1)*(1-x^2)^n, for n>=0; e.g., C_n(x) = R_n(x)/(1-x)^(n+1)/(1-x^2)^n.
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 18, 10, 1, 1, 15, 43, 43, 15, 1, 1, 21, 86, 135, 87, 21, 1, 1, 28, 156, 345, 345, 159, 28, 1, 1, 36, 260, 771, 1083, 777, 267, 36, 1, 1, 45, 410, 1557, 2901, 2927, 1577, 423, 45, 1, 1, 55, 615, 2913, 6909, 9219, 7001, 2973, 637, 55, 1
Offset: 0
Triangle begins:
1;
1,1;
1,3,1;
1,6,6,1;
1,10,18,10,1;
1,15,43,43,15,1;
1,21,86,135,87,21,1;
1,28,156,345,345,159,28,1;
1,36,260,771,1083,777,267,36,1;
1,45,410,1557,2901,2927,1577,423,45,1;
1,55,615,2913,6909,9219,7001,2973,637,55,1; ...
where g.f. for columns is formed from g.f. of rows:
column 2: (1 + 3*x + 1*x^2)/(1-x)^3/(1-x^2)^2 = 1 + 6*x + 18*x^2 + 43*x^3 + 86*x^4 + 156*x^5 +...
column 3: (1 + 6*x + 6*x^2 + 1*x^3)/(1-x)^4/(1-x^2)^3 = 1 + 10*x + 43*x^2 + 135*x^3 + 345*x^4 + 771*x^5 +...
column 4: (1 + 10*x + 18*x^2 + 10*x^3 + 1*x^4)/(1-x)^5/(1-x^2)^4 = 1 + 15*x + 87*x^2 + 345*x^3 + 1083*x^4 + 2901*x^5 +...
-
T(n,k)=if(n
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{T(n,k)=if(n==k,1,sum(j=0,k,T(k,j)*sum(i=0,n-j-k, (-1)^(n-i-j-k)*binomial(2*k+i,i)*binomial(n-i-j-1,n-i-j-k))))} \\ Paul D. Hanna, Jun 21 2006
Showing 1-4 of 4 results.