A300401 Array T(n,k) = n*(binomial(k, 2) + 1) + k*(binomial(n, 2) + 1) read by antidiagonals.
0, 1, 1, 2, 2, 2, 3, 4, 4, 3, 4, 7, 8, 7, 4, 5, 11, 14, 14, 11, 5, 6, 16, 22, 24, 22, 16, 6, 7, 22, 32, 37, 37, 32, 22, 7, 8, 29, 44, 53, 56, 53, 44, 29, 8, 9, 37, 58, 72, 79, 79, 72, 58, 37, 9, 10, 46, 74, 94, 106, 110, 106, 94, 74, 46, 10, 11, 56, 92, 119
Offset: 0
Examples
The array T(n,k) begins 0 1 2 3 4 5 6 7 8 9 10 11 ... 1 2 4 7 11 16 22 29 37 46 56 67 ... 2 4 8 14 22 32 44 58 74 92 112 134 ... 3 7 14 24 37 53 72 94 119 147 178 212 ... 4 11 22 37 56 79 106 137 172 211 254 301 ... 5 16 32 53 79 110 146 187 233 284 340 401 ... 6 22 44 72 106 146 192 244 302 366 436 512 ... 7 29 58 94 137 187 244 308 379 457 542 634 ... 8 37 74 119 172 233 302 379 464 557 658 767 ... 9 46 92 147 211 284 366 457 557 666 784 911 ... 10 56 112 178 254 340 436 542 658 784 920 1066 ... 11 67 134 212 301 401 512 634 767 911 1066 1232 ... 12 79 158 249 352 467 594 733 884 1047 1222 1409 ... 13 92 184 289 407 538 682 839 1009 1192 1388 1597 ... 14 106 212 332 466 614 776 952 1142 1346 1564 1796 ... 15 121 242 378 529 695 876 1072 1283 1509 1750 2006 ... 16 137 274 427 596 781 982 1199 1432 1681 1946 2227 ... 17 154 308 479 667 872 1094 1333 1589 1862 2152 2459 ... 18 172 344 534 742 968 1212 1474 1754 2052 2368 2702 ... 19 191 382 592 821 1069 1336 1622 1927 2251 2594 2956 ... 20 211 422 653 904 1175 1466 1777 2108 2459 2830 3221 ... ... The inverse binomial transforms of the columns are 0 1 2 3 4 5 6 7 8 9 10 11 ... A001477 1 1 2 4 7 11 22 29 37 45 56 67 ... A152947 0 1 2 3 4 5 6 7 8 9 10 11 ... A001477 0 0 0 0 0 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 0 0 0 ... ...
References
- Miklós Bóna, Introduction to Enumerative Combinatorics, McGraw-Hill, 2007.
- L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Reidel Publishing Company, 1974.
- R. P. Stanley, Enumerative Combinatorics, second edition, Cambridge University Press, 2011.
Links
- Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
- Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657 [math.CO], 2014.
- Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
Crossrefs
Programs
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Maple
T := (n, k) -> n*(binomial(k, 2) + 1) + k*(binomial(n, 2) + 1); for n from 0 to 20 do seq(T(n, k), k = 0 .. 20) od;
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Mathematica
T[n_, k_] := n (Binomial[k, 2] + 1) + k (Binomial[n, 2] + 1); Table[T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 07 2018 *) -
Maxima
T(n, k) := n*(binomial(k, 2) + 1) + k*(binomial(n, 2) + 1)$ for n:0 thru 20 do print(makelist(T(n, k), k, 0, 20));
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PARI
T(n, k) = n*(binomial(k,2) + 1) + k*(binomial(n,2) + 1); tabl(nn) = for (n=0, nn, for (k=0, nn, print1(T(n, k), ", ")); print); \\ Michel Marcus, Mar 12 2018
Formula
T(n,0) = A001477(n).
T(n,1) = A000124(n).
T(n,2) = A014206(n).
T(n,3) = A273465(3*n+2).
T(n,4) = A084849(n+1).
T(n,n) = A179000(n-1,n), n >= 1.
T(2*n,2*n) = 8*A081436(n-1), n >= 1.
T(2*n+1,2*n+1) = 2*A006000(2*n+1).
T(n,n+1) = A188377(n+3).
T(n,n+2) = A188377(n+2), n >= 1.
Sum_{k=0..n} T(k,n-k) = 2*(binomial(n, 4) + binomial(n, 2)).
G.f.: -((2*x*y - y - x)*(2*x*y - y - x + 1))/(((x - 1)*(y - 1))^3).
E.g.f.: (1/2)*(x + y)*(x*y + 2)*exp(x + y).
Comments