A013612 → x ↦ x+1 → reverse

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Submission text ?

Triangle begins:
  0:       25,        60,         36;
  1:      125,       450,        540,         216;
  2:      625,      3000,       5400,        4320,        1296;
  3:     3125,     18750,      45000,       54000,       32400,        7776;
  4:    15625,    112500,     337500,      540000,      486000,      233280,       46656;
  5:    78125,    656250,    2362500,     4725000,     5670000,     4082400,     1632960,      279936;
  6:   390625,   3750000,   15750000,    37800000,    56700000,    54432000,    32659200,    11197440,     1679616;
  7:  1953125,  21093750,  101250000,   283500000,   510300000,   612360000,   489888000,   251942400,    75582720,    10077696;
  8:  9765625, 117187500,  632812500,  2025000000,  4252500000,  6123600000,  6123600000,  4199040000,  1889568000,   503884800,   60466176;
  9: 48828125, 644531250, 3867187500, 13921875000, 33412500000, 56133000000, 67359600000, 57736800000, 34642080000, 13856832000, 3325639680, 362797056;

The row polynomials satisfy P(n,x) = (5+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 25 + 60*x + 36*x^2.

P[0, x_] := 25 + 60*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[(5 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 25 + 60*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (5 + 6*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 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)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A007318 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 Triangle of coefficients in expansion of (1+5x)^n.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A038336 → D
• A038329 → x ↦ x+1 → x ↦ x/(x+1)
• A038255 → basis A013610 → basis A013611
• A038271 → basis A013609 → D
• A038220 → basis A013610 → D
• A038271 → basis A013609
• A038338 → basis A013610 → basis A013609
• A147716 → basis A013610 → x ↦ x/(x+1)
• A038334 → basis A013610 → reverse
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.
• A038220 → basis A013610
• A178822 → basis A013610 → basis A013609
• A085880 → basis A013610 → basis A013609
• A038314 → D
• A013612 → x ↦ x+1 → D
• A013612 → x ↦ x+1
• A038259 → D
• A038221 → basis A013610 → basis A013609
• A196347 → basis A013610 → basis A013609
• A178820 → basis A013610 → basis A013609
• A038334 → basis A013610 → D
• A038334 → basis A013610
• A178821 → basis A013610 → basis A013609
• A028326 → basis A013610 → basis A013609
• A038259: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.
• A120910 → basis A013609 → basis A013610
• Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.
• A038291 → basis A013609 → x ↦ x/(x+1)
• A156992 → 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.