A038338 → basis A013610 → x ↦ x+1

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Submission text ?

Triangle begins:
  0:                 3600,                  4320,                  1296;
  1:               216000,                388800,                233280,                  46656;
  2:             12960000,              31104000,              27993600,               11197440,                1679616;
  3:            777600000,            2332800000,            2799360000,             1679616000,              503884800,               60466176;
  4:          46656000000,          167961600000,          251942400000,           201553920000,            90699264000,            21767823360,            2176782336;
  5:        2799360000000,        11757312000000,        21163161600000,         21163161600000,         12697896960000,          4571242905600,          914248581120,           78364164096;
  6:      167961600000000,       806215680000000,      1693052928000000,       2031663513600000,       1523747635200000,        731398864896000,       219419659468800,        37614798766080,         2821109907456;
  7:    10077696000000000,     54419558400000000,    130606940160000000,     182849716224000000,     164564744601600000,      98738846760960000,     39495538704384000,     10155995666841600,      1523399350026240,      101559956668416;
  8:   604661760000000000,   3627970560000000000,   9795520512000000000,   15672832819200000000,   16456474460160000000,   11848661611315200000,   5924330805657600000,   2031199133368320000,    457019805007872000,    60935974001049600,    3656158440062976;
  9: 36279705600000000000, 239446056960000000000, 718338170880000000000, 1293008707584000000000, 1551610449100800000000, 1303352777244672000000, 782011666346803200000, 335147857005772800000, 100544357101731840000, 20108871420346368000, 2413064570441564160, 131621703842267136;

The row polynomials satisfy P(n,x) = (60+36*x)*P(n-1,x) for n >= 1, with P(0,x) = 3600 + 4320*x + 1296*x^2.

P[0, x_] := 3600 + 4320*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[(60 + 36*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 3600 + 4320*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = (60 + 36*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 whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^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 Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
    • ↑ Geometric basis map (elementary) (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A038243 → basis A013611 → x ↦ x/(x+1)
• A178820 → basis A013610 → x ↦ x+1
• A038271 → basis A013610 → basis A013610
• A178821 → basis A013610 → x ↦ x+1
• A013622 → basis A013609 → basis A013610
• A038329 → basis A013610
• A028326 → basis A013610 → x ↦ x+1
• A038329 → basis A013610 → reverse
• A027467 → basis A013611 → basis A013610
• A038271 → basis A013611 → D
• A178822 → basis A013610 → basis A007318
• A038271 → basis A013611
• A013620 → basis A007318
• A013612 → basis A013609 → D
• A013622: Triangle of coefficients in expansion of (3+5x)^n.
• A038329 → basis A013610 → D
• A038221 → basis A013610 → basis A007318
• A038319 → basis A013611 → basis A013610
• A038231 → D → basis A013610
• A013622 → D
• A013620 → basis A007318 → D
• A196347 → basis A013610 → basis A007318
• A038284 → basis A007318 → basis A013609
• A013612 → basis A013609 → reverse
• A038334 → x ↦ x/(x+1)
• A156992 → basis A013610 → x ↦ x+1
• A038255 → basis A013610 → basis A013609
• A013612 → basis A013609
• A120910 → basis A013609 → x ↦ x/(x+1)
• A038335 → x ↦ x+1 → basis A013609
• A085880 → basis A013610 → 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.