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 +...
A114174
Central terms of triangle A114172 found in even-indexed rows.
Original entry on oeis.org
1, 3, 16, 95, 601, 3951, 26650, 183043, 1274080, 8959268, 63512890, 453227590, 3252065077, 23443729683, 169680654790, 1232381601759, 8977925594880, 65579423363511, 480162707557696, 3523121247341469, 25899477669429042
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.