A013618 → basis A013610 → x ↦ x+1

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Submission text ?

Triangle begins:
  0:                2025,                 2970,                 1089;
  1:               91125,               200475,               147015,                35937;
  2:             4100625,             12028500,             13231350,              6468660,               1185921;
  3:           184528125,            676603125,            992351250,            727724250,             266832225,              39135393;
  4:          8303765625,          36536568750,          66983709375,          65495182500,           36022350375,           10566556110,            1291467969;
  5:        373669453125,        1918169859375,        4219973690625,        5157745621875,         3782346789375,         1664232587325,          406812410235,          42618442977;
  6:      16815125390625,       98648735625000,      253198421437500,      371357684775000,       340411211043750,       199707910479000,        73226233842300,       15342639471720,        1406408618241;
  7:     756680642578125,     4994092241015625,    14649337240312500,    25066643722312500,     27573308094543750,     20220425935998750,      9885541568710500,     3106884493023300,      569595490387605,      46411484401953;
  8:   34050628916015625,   249704612050781250,   824025219767578125,  1611427096434375000,   2067998107090781250,   1819838334239887500,   1112123426479931250,   466032673953495000,   128158985337211125,   20885167980878850,   1531578985264449;
  9: 1532278301220703125, 12360378296513671875, 45321387087216796875, 99707051591876953125, 146237009001419531250, 150136662574790718750, 110100219221513193750, 57671543401745006250, 21146232580639835625, 5169079075267515375, 758131597705902255, 50542106513726817;

The row polynomials satisfy P(n,x) = (45+33*x)*P(n-1,x) for n >= 1, with P(0,x) = 2025 + 2970*x + 1089*x^2.

P[0, x_] := 2025 + 2970*x + 1089*x^2;
P[n_, x_] := P[n, x] = Expand[(45 + 33*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 2025 + 2970*x + 1089*x^2; for(n = 1, N, my(i = n + 1); v[i] = (45 + 33*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 Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (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 of coefficients in expansion of (1+11x)^n.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of coefficients in expansion of (1+11x)^n.?
        • 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 of coefficients in expansion of (1+11x)^n.?
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A038259 → basis A013610
• A038259 → basis A013610 → D
• A038271 → basis A013609 → basis A013610
• A013612 → x ↦ x+1 → basis A013610
• A038336 → basis A013610
• A038259 → basis A013610 → reverse

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.