A008955 → basis A007318 → reverse

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Submission text ?

Triangle begins:
  0:            1;
  1:            1,             2;
  2:            4,            13,            10;
  3:           36,           157,           220,            100;
  4:          576,          3124,          6189,           5340,           1700;
  5:        14400,         93076,        235949,         294414,         181340,          44200;
  6:       518400,       3883536,      11937976,       19329017,       17421558,        8300780,        1635400;
  7:     25401600,     216213264,     779137624,     1544020633,     1820107192,     1277816120,      495173600,       81770000;
  8:   1625702400,   15488752896,   63918670096,   149461266072,   216848201433,   200087199160,   114749158200,    37419564000,    5315050000;
  9: 131681894400, 1387896581376, 6447490015248, 17347693499704, 29820528133977, 33988615649466, 25701832145320, 12440415656400, 3498923298000, 435834100000;

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

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n^2 + (n^2 + 1)*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] = (n^2 + (n^2 + 1)*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 Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A007318 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • 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
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A008955 → basis A007318
• A269944 → x ↦ x/(x+1)

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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.