A038321 → basis A013610 → basis A007318

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Submission text ?

Triangle begins:
  0:               1521,                1638,                 441;
  1:              59319,               95823,               51597,                9261;
  2:            2313441,             4982796,             4024566,             1444716,              194481;
  3:           90224199,           242911305,           261596790,           140859810,            37923795,             4084101;
  4:         3518743761,         11368249074,         15303412215,         10987065180,          4437084015,           955679634,            85766121;
  5:       137231006679,        517255332867,        835566306939,        749867198535,        403774645365,        130450270041,         23414151033,          1801088541;
  6:      5352009260481,      23054809122072,      43449447960828,      46791713188584,      31494422338470,      13566828084264,       3652607561148,        561939624792,        37822859361;
  7:    208728361158759,    1011529750230909,    2178679462035804,    2737315221532164,    2210908448160594,    1190489164394166,     427355084654316,      98620404150996,     13275823635711,      794280046581;
  8:   8140406085191601,   43832955843339390,  106210623774245445,  152507562342506280,  143709049130438610,   92858154822744948,   41667120753795810,   12820652539629480,   2588785608963645,   309769218166590,   16679880978201;
  9: 317475837322472439, 1880433805679259831, 5062706399905699545, 8178218030616899265, 8807311725279737670, 6639358069826263782, 3575038960675680498, 1375014984875261730, 370196342081801235, 66445497296733555, 7155668939648229, 350277500542221;

The row polynomials satisfy P(n,x) = (39+21*x)*P(n-1,x) for n >= 1, with P(0,x) = 1521 + 1638*x + 441*x^2.

P[0, x_] := 1521 + 1638*x + 441*x^2;
P[n_, x_] := P[n, x] = Expand[(39 + 21*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1521 + 1638*x + 441*x^2; for(n = 1, N, my(i = n + 1); v[i] = (39 + 21*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_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^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_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A013613 → x ↦ x+1 → x ↦ x/(x+1)
• A038335 → basis A013609 → x ↦ x/(x+1)
• A038333 → x ↦ x+1 → basis A013609
• A038284 → basis A013609 → 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.