A143491 → basis A013611 → basis A013610

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:           56,           180,            144;
  1:          504,          2292,           3456,            1728;
  2:         5040,         28968,          62064,           58752,           20736;
  3:        55440,        379128,        1030320,         1391040,          933120,          248832;
  4:       665280,       5214816,       16913376,        29056320,        27889920,        14183424,         2985984;
  5:      8648640,      75775968,      282451680,       580692672,       711244800,       519063552,       209018880,        35831808;
  6:    121080960,    1164647232,     4863635136,     11519117568,     16925739264,     15801827328,      9155026944,      3009871872,       429981696;
  7:   1816214400,   18922680000,    86930293824,    231150385152,    392115499776,    440136281088,    326947332096,    155008401408,     42568187904,     5159780352;
  8:  29059430400,  324557452800,  1617956861184,   4741569688320,   9047652618240,  11747566494720,  10512792686592,   6403502407680,   2541191823360,   593374740480,   61917364224;
  9: 494010316800, 5866189862400, 31399956073728, 100022167035648, 210708930769920, 308280461829120, 319688273608704, 235013053169664, 120042289889280, 40581672468480, 8173092077568, 743008370688;

The row polynomials satisfy P(n,x) = (n+8+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 56 + 180*x + 144*x^2.

P[0, x_] := 56 + 180*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(n + 8 + 12*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 56 + 180*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (n + 8 + 12*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_A013611 of Unsigned 2-Stirling numbers of the first kind.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Unsigned 2-Stirling numbers of the first kind.?
        • 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 Unsigned 2-Stirling numbers of the first kind.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A136124 → basis A013611 → basis A013610

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.