A038226 → basis A013610 → D

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Submission text ?

Triangle begins:
  0:           528,            1152;
  1:          8712,           38016,            41472;
  2:        127776,          836352,          1824768,          1327104;
  3:       1756920,        15333120,         50181120,         72990720,          39813120;
  4:      23191344,       252996480,       1103984640,       2408693760,        2627665920,        1146617856;
  5:     297622248,      3896145792,      21251704320,      61823139840,      101165137920,       88289574912,        32105299968;
  6:    3741536832,     57143471616,     374029996032,    1360109076480,     2967510712320,     3884741296128,      2825266397184,       880602513408;
  7:   46301518296,    808171955712,    6171494934528,   26930159714304,    73445890129920,   128196462772224,    139850686660608,     87179648827392,     23776267862016;
  8:  565907445840,  11112364391040,   96980634685440,  493719594762240,  1615809582858240,  3525402726236160,   5127858510888960,   4794880685506560,   2615389464821760,   634033809653760;
  9: 6847480094664, 149399565701760, 1466832099617280, 8534295852318720, 32585493254307840, 85314745974915072, 155117719954391040, 193393520982097920, 158231062621716480, 76718090968104960, 16738492574859264;

The row polynomials satisfy (n+1)*P(n,x) = (11*n+22+(24*n+48)*x)*P(n-1,x) for n >= 1, with P(0,x) = 528 + 1152*x.

P[0, x_] := 528 + 1152*x;
P[n_, x_] := P[n, x] = Expand[Together[((11*n + 22 + (24*n + 48)*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] = 528 + 1152*x; for(n = 1, N, my(i = n + 1); v[i] = ((11*n + 22 + (24*n + 48)*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 basis_map_A013610 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^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
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.?
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Liu–Wang theorem (above)
                • 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
            • 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
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A038226 → basis A013610
• A038226 → basis A013610 → reverse
• A013622 → x ↦ x/(x+1) → basis A013610
• A038314 → basis A013611

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.