A013612 → reverse

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Submission text ?

Triangle begins:
  0:       25,        10,         1;
  1:      125,        75,        15,        1;
  2:      625,       500,       150,       20,        1;
  3:     3125,      3125,      1250,      250,       25,       1;
  4:    15625,     18750,      9375,     2500,      375,      30,       1;
  5:    78125,    109375,     65625,    21875,     4375,     525,      35,      1;
  6:   390625,    625000,    437500,   175000,    43750,    7000,     700,     40,     1;
  7:  1953125,   3515625,   2812500,  1312500,   393750,   78750,   10500,    900,    45,    1;
  8:  9765625,  19531250,  17578125,  9375000,  3281250,  787500,  131250,  15000,  1125,   50,  1;
  9: 48828125, 107421875, 107421875, 64453125, 25781250, 7218750, 1443750, 206250, 20625, 1375, 55, 1;

The row polynomials satisfy P(n,x) = (5+x)*P(n-1,x) for n >= 1, with P(0,x) = 25 + 10*x + x^2.

P[0, x_] := 25 + 10*x + x^2;
P[n_, x_] := P[n, x] = Expand[(5 + 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 + 10*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (5 + 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 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

Related ?

Same fingerprint ?

• A013611 → x ↦ x/(x+1)
• A038329 → x ↦ x+1 → reverse
• A038330 → basis A007318 → basis A007318
• A038329 → x ↦ x+1 → D
• A013612 → D
• A147716 → basis A013610 → reverse
• A027467 → basis A013611 → x ↦ x+1
• A038243 → reverse
• basis-map coefficients of A112227 → basis A013611 → x ↦ x/(x+1)
• A147716 → basis A013610 → D
• A038243: Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).
• A136572 → basis A013611 → x ↦ x/(x+1)
• A038291 → basis A013609
• A038243 → D
• Triangle of coefficients in expansion of (1+5x)^n.
• A147716 → basis A013610
• A130123 → basis A013611 → x ↦ x/(x+1)
• A038291 → basis A013609 → reverse
• A038291 → basis A013609 → D
• A038329 → x ↦ x+1
• basis-map coefficients of A026729 → basis A013611 → 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.