A204420 → basis A013609 → basis A013610

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Submission text ?

Triangle begins:
  0:                 1;
  1:                 3,                  6;
  2:                45,                144,                108;
  3:              1575,               6390,               8100,               3240;
  4:             99225,             468720,             778680,             544320,             136080;
  5:           9823275,           51761430,          102400200,           95936400,           42865200,            7348320;
  6:        1404728325,         8050220640,        18059482980,        20477318400,        12461526000,         3879912960,          484989120;
  7:      273922023375,      1679361834150,      4149516391020,      5401716760440,      4027228405200,      1728582055200,       397206089280,       37829151360;
  8:    69850115960625,    452890249812000,   1209269244783600,   1750894249104000,   1513097751765600,    803238980544000,    256859937734400,    45394981632000,    3404623622400;
  9: 22561587455281875, 153408262517259750, 436788771545926800, 688884305428519200, 667321787228896800, 413782161395803200, 164896135903699200, 40862292716044800, 5729981556499200, 347271609484800;

The row polynomials satisfy P(n,x) = (4*n^2-1+(12*n-6)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4*n^2 - 1 + (12*n - 6)*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (4*n^2 - 1 + (12*n - 6)*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013609 of Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of e…?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.?
        • Parent-row provers of polytool polytool ? step › direct check
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.?
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A290319 → basis A013611 → basis A013610
• A161198 → basis A013610
• A161198 → basis A013610 → reverse

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.