A013612 → basis A013610 → basis A013609

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Submission text ?

Triangle begins:
  0:             441,             1260,               900;
  1:            9261,            39690,             56700,              27000;
  2:          194481,          1111320,           2381400,            2268000,             810000;
  3:         4084101,         29172150,          83349000,          119070000,           85050000,           24300000;
  4:        85766121,        735138180,        2625493500,         5000940000,         5358150000,         3061800000,           729000000;
  5:      1801088541,      18010885410,       77189508900,       183784545000,       262549350000,       225042300000,        107163000000,         21870000000;
  6:     37822859361,     432261249840,     2161306249200,      6175160712000,     11027072700000,     12602368800000,       9001692000000,       3674160000000,        656100000000;
  7:    794280046581,   10212172027470,    58355268728400,    194517562428000,    416823348060000,    595461925800000,     567106596000000,     347208120000000,     124002900000000,     19683000000000;
  8:  16679880978201,  238284013974300,  1531825804120500,   5835526872840000,  14588817182100000,  25009400883600000,   29773096290000000,   24304568400000000,   13020304500000000,   4133430000000000,    590490000000000;
  9: 350277500542221, 5504360722806330, 39316862305759500, 168500838453255000, 481430967009300000, 962861934018600000, 1375517048598000000, 1403588825100000000, 1002563446500000000, 477411165000000000, 136403190000000000, 17714700000000000;

The row polynomials satisfy P(n,x) = (21+30*x)*P(n-1,x) for n >= 1, with P(0,x) = 441 + 1260*x + 900*x^2.

P[0, x_] := 441 + 1260*x + 900*x^2;
P[n_, x_] := P[n, x] = Expand[(21 + 30*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 441 + 1260*x + 900*x^2; for(n = 1, N, my(i = n + 1); v[i] = (21 + 30*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)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013610 of Triangle of coefficients in expansion of (1+5x)^n.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of coefficients in expansion of (1+5x)^n.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? 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
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis_map_A013610 of Triangle of coefficients in expansion of (1+5x)^n.?
            • ↑ Geometric basis map (elementary) (above)
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
            • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
            • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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

Related ?

Same fingerprint ?

• A038284 → basis A007318 → x ↦ x/(x+1)
• A013622 → basis A013611 → basis A013609
• A038324 → basis A013610
• A038255 → basis A013610 → x ↦ x/(x+1)
• A038336 → basis A013609 → 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.