A013612 → D

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Submission text ?

Triangle begins:
  0: 10,   50;
  1: 15,  150,   375;
  2: 20,  300,  1500,   2500;
  3: 25,  500,  3750,  12500,   15625;
  4: 30,  750,  7500,  37500,   93750,    93750;
  5: 35, 1050, 13125,  87500,  328125,   656250,    546875;
  6: 40, 1400, 21000, 175000,  875000,  2625000,   4375000,   3125000;
  7: 45, 1800, 31500, 315000, 1968750,  7875000,  19687500,  28125000,  17578125;
  8: 50, 2250, 45000, 525000, 3937500, 19687500,  65625000, 140625000, 175781250,   97656250;
  9: 55, 2750, 61875, 825000, 7218750, 43312500, 180468750, 515625000, 966796875, 1074218750, 537109375;

The row polynomials satisfy (n+1)*P(n,x) = (n+2+(5*n+10)*x)*P(n-1,x) for n >= 1, with P(0,x) = 10 + 50*x.

P[0, x_] := 10 + 50*x;
P[n_, x_] := P[n, x] = Expand[Together[((n + 2 + (5*n + 10)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 10 + 50*x; for(n = 1, N, my(i = n + 1); v[i] = ((n + 2 + (5*n + 10)*x)*v[i - 1])/(n + 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 Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] 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 ?

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