A013612 → x ↦ x+1 → basis A013609

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Submission text ?

Triangle begins:
  0:          121,           220,            100;
  1:         1331,          3630,           3300,           1000;
  2:        14641,         53240,          72600,          44000,          10000;
  3:       161051,        732050,        1331000,        1210000,         550000,         100000;
  4:      1771561,       9663060,       21961500,       26620000,       18150000,        6600000,        1000000;
  5:     19487171,     124009270,      338207100,      512435000,      465850000,      254100000,       77000000,       10000000;
  6:    214358881,    1558973680,     4960370800,     9018856000,    10248700000,     7453600000,     3388000000,      880000000,      100000000;
  7:   2357947691,   19292299290,    70153815600,   148811124000,   202924260000,   184476600000,   111804000000,    43560000000,     9900000000,    1000000000;
  8:  25937424601,  235794769100,   964614964500,  2338460520000,  3720278100000,  4058485200000,  3074610000000,  1597200000000,   544500000000,  110000000000,   10000000000;
  9: 285311670611, 2853116706110, 12968712300500, 35369215365000, 64307664300000, 81846118200000, 74405562000000, 48315300000000, 21961500000000, 6655000000000, 1210000000000, 100000000000;

The row polynomials satisfy P(n,x) = (11+10*x)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 220*x + 100*x^2.

P[0, x_] := 121 + 220*x + 100*x^2;
P[n_, x_] := P[n, x] = Expand[(11 + 10*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 121 + 220*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = (11 + 10*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 Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (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 → basis A013609
• A038324 → D
• A038277 → basis A013609 → basis A013609
• A038301 → basis A013609 → D
• A038324 → reverse
• A038336 → basis A013609 → D
• A038301 → basis A013609
• A038303 → x ↦ x/(x+1)
• A038324: Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^j.
• A038271 → basis A013609 → 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.