A038284 → basis A013609 → basis A007318

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Submission text ?

Triangle begins:
  0:              676,               624,               144;
  1:            17576,             24336,             11232,              1728;
  2:           456976,            843648,            584064,            179712,             20736;
  3:         11881376,          27418560,          25309440,          11681280,           2695680,            248832;
  4:        308915776,         855459072,         987068160,         607426560,         210263040,          38817792,           2985984;
  5:       8031810176,       25948925184,       35929281024,       27637908480,       12755957760,        3532419072,         543449088,         35831808;
  6:     208827064576,      771053776896,     1245548408832,     1149736992768,      663309803520,      244914388992,       56518705152,       7453016064,        429981696;
  7:    5429503678976,    22553322974208,    41636903952384,    44839742717952,    31042898804736,    14327491756032,     4408459001856,     872002879488,     100615716864,      5159780352;
  8:  141167095653376,   651540441477120,  1353199378452480,  1665476158095360,  1345192281538560,   745029571313664,   286549835120640,   75573582888960,   13080043192320,   1341542891520,    61917364224;
  9: 3670344486987776, 18634056626245632, 43001669137489920, 59540772651909120, 54960713217146880, 35513076232617984, 16390650568900608, 5403511176560640, 1246964117667840, 191840633487360, 17708366168064, 743008370688;

The row polynomials satisfy P(n,x) = (26+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 676 + 624*x + 144*x^2.

P[0, x_] := 676 + 624*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(26 + 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] = 676 + 624*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (26 + 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 (−∞, -1]
  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 Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Affine, reversal and Möbius substitutions (elementary)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^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
        • 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)*8^(i-j)*6^j.?
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A038321 → basis A013610 → x ↦ x/(x+1)
• A038305 → basis A013609 → reverse
• A038335 → basis A013609 → x ↦ x+1
• A038305 → basis A013609 → D
• A038305 → basis A013609
• A038291 → basis A013610 → basis A013609

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.