A322229 Triangle, read by rows, where row n lists coefficients in Product_{k=1..n} (k + x + (n+1-k)*x^2), for n >= 0.

1, 1, 1, 1, 2, 3, 6, 3, 2, 6, 11, 32, 27, 32, 11, 6, 24, 50, 189, 210, 350, 210, 189, 50, 24, 120, 274, 1269, 1689, 3594, 2915, 3594, 1689, 1269, 274, 120, 720, 1764, 9652, 14651, 37750, 37457, 58156, 37457, 37750, 14651, 9652, 1764, 720, 5040, 13068, 82396, 138473, 417780, 481074, 896412, 714483, 896412, 481074, 417780, 138473, 82396, 13068, 5040, 40320, 109584, 781820, 1426428, 4923585, 6370164, 13808832, 12899520, 19279494, 12899520, 13808832, 6370164, 4923585, 1426428, 781820, 109584, 40320

Offset

0,5

Comments

Row sums equal A000272(n+2) = (n+2)^n, for n >= 0.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1680 terms of this irregular triangle, as read by rows 0..40.

Formula

T(n,0) = T(n,2*n) = n!, for n >= 0.

Sum_{k=0..2*n} T(n,k) = (n+2)^n, for n >= 0.

Sum_{k=0..2*n} T(n,k)*(-1)^k = n^n, for n >= 0.

Example

This triangle, where row n gives coefficients in Product_{k=1..n} (k + x + (n+1-k)*x^2), begins
1;
1, 1, 1;
2, 3, 6, 3, 2;
6, 11, 32, 27, 32, 11, 6;
24, 50, 189, 210, 350, 210, 189, 50, 24;
120, 274, 1269, 1689, 3594, 2915, 3594, 1689, 1269, 274, 120;
720, 1764, 9652, 14651, 37750, 37457, 58156, 37457, 37750, 14651, 9652, 1764, 720;
5040, 13068, 82396, 138473, 417780, 481074, 896412, 714483, 896412, 481074, 417780, 138473, 82396, 13068, 5040;
40320, 109584, 781820, 1426428, 4923585, 6370164, 13808832, 12899520, 19279494, 12899520, 13808832, 6370164, 4923585, 1426428, 781820, 109584, 40320;
362880, 1026576, 8172540, 15965072, 61978425, 88164321, 217535135, 230299722, 398293065, 314352219, 398293065, 230299722, 217535135, 88164321, 61978425, 15965072, 8172540, 1026576, 362880; ...
Example of row generating functions.
Row 0: 1;
Row 1: (1 + x + 1*x^2);
Row 2: (1 + x + 2*x^2)*(2 + x + 1*x^2)  =  2 + 3*x + 6*x^2 + 3*x^3 + 2*x^4;
Row 3: (1 + x + 3*x^2)*(2 + x + 2*x^2)*(3 + x + 1*x^2)  =  6 + 11*x + 32*x^2 + 27*x^3 + 32*x^4 + 11*x^5 + 6*x^6;
Row 4: (1 + x + 4*x^2)*(2 + x + 3*x^2)*(3 + x + 2*x^2)*(4 + x + 1*x^2)  =  24 + 50*x + 189*x^2 + 210*x^3 + 350*x^4 + 210*x^5 + 189*x^6 + 50*x^7 + 24*x^8;
Row 5: (1 + x + 5*x^2)*(2 + x + 4*x^2)*(3 + x + 3*x^2)*(4 + x + 2*x^2)*(5 + x + 1*x^2)  =  120 + 274*x + 1269*x^2 + 1689*x^3 + 3594*x^4 + 2915*x^5 + 3594*x^6 + 1689*x^7 + 1269*x^8 + 274*x^9 + 120*x^10;
...
Row sums = [1, 3, 16, 125, 1296, 16807, 262144, 4782969, ..., (n+2)^n, ...].
Main diagonal = [1, 1, 6, 27, 350, 2915, 58156, 714483, ..., A322233(n), ...].
Secondary diagonal = [1, 3, 32, 210, 3594, 37457, 896412, ..., A322234(n), ...].

Mathematica

row[n_] := Product[k + x + (n - k + 1) x^2, {k, 1, n}] + O[x]^(2 n + 1) // CoefficientList[#, x]&;
Table[row[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Dec 29 2018 *)

Prog

(PARI) {T(n, k) = polcoeff( prod(m=1, n, m + x + (n+1-m)*x^2) +x*O(x^k), k)}
for(n=0, 10, for(k=0, 2*n, print1( T(n, k), ", ")); print(""))

Crossrefs

Cf. A000272 (row sums), A322233 (main diagonal), A322234 (diagonal).

Sequence in context: A189971 A211605 A316610 * A245886 A224504 A216059

Adjacent sequences: A322226 A322227 A322228 * A322230 A322231 A322232

Keyword

nonn,tabf

Author

Paul D. Hanna, Dec 18 2018

Status

approved