A038271 → reverse

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Submission text ?

Triangle begins:
  0:       1;
  1:       5,        7;
  2:      25,       70,        49;
  3:     125,      525,       735,       343;
  4:     625,     3500,      7350,      6860,      2401;
  5:    3125,    21875,     61250,     85750,     60025,      16807;
  6:   15625,   131250,    459375,    857500,    900375,     504210,     117649;
  7:   78125,   765625,   3215625,   7503125,  10504375,    8823675,    4117715,    823543;
  8:  390625,  4375000,  21437500,  60025000, 105043750,  117649000,   82354300,  32941720,   5764801;
  9: 1953125, 24609375, 137812500, 450187500, 945393750, 1323551250, 1235314500, 741188700, 259416045, 40353607;

The row polynomials satisfy P(n,x) = (5+7*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(5 + 7*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (5 + 7*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 Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^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
    • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    • Parent-row provers of polytool polytool ? step › direct check

Related ?

Candidate equalities ?

• A038220 → basis A007318 → x ↦ x/(x+1) 15 rows; no all-index equality certificate was found

Same fingerprint ?

• A038334 → x ↦ x+1 → x ↦ x/(x+1)
• A038271 → D
• A038329 → basis A013609 → x ↦ x/(x+1)
• A038271: Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.
• A038291 → basis A013611 → x ↦ x/(x+1)
• A013622 → basis A013611 → basis A007318
• A013612 → basis A013610 → basis A007318

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.