A038314 → x ↦ x/(x+1)

Matches ?

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Submission text ?

Triangle begins:
  0:          100,           440,            484;
  1:         1000,          6600,          14520,           10648;
  2:        10000,         88000,         290400,          425920,          234256;
  3:       100000,       1100000,        4840000,        10648000,        11712800,          5153632;
  4:      1000000,      13200000,       72600000,       212960000,       351384000,        309217920,        113379904;
  5:     10000000,     154000000,     1016400000,      3726800000,      8198960000,      10822627200,       7936593280,       2494357888;
  6:    100000000,    1760000000,    13552000000,     59628800000,    163979200000,     288603392000,     317463731200,     199548631040,      54875873536;
  7:   1000000000,   19800000000,   174240000000,    894432000000,   2951625600000,    6493576320000,    9523911936000,    8979688396800,    4938828618240,    1207269217792;
  8:  10000000000,  220000000000,  2178000000000,  12777600000000,  49193760000000,  129871526400000,  238097798400000,  299322946560000,  246941430912000,  120726921779200,   26559922791424;
  9: 100000000000, 2420000000000, 26620000000000, 175692000000000, 773044800000000, 2380977984000000, 5238151564800000, 8231381030400000, 9054519133440000, 6639980697856000, 2921591507056640, 584318301411328;

The row polynomials satisfy (1+x)*P(n,x) = (10+32*x+22*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 100 + 440*x + 484*x^2.

P[0, x_] := 100 + 440*x + 484*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((10 + 32*x + 22*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 100 + 440*x + 484*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((10 + 32*x + 22*x^2)*v[i - 1])/(1 + x)); 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 [-1, 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)*10^(i-j)*12^j.?
    • 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 whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.?
        • 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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.?
            • ↑ Liu–Wang theorem (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Candidate equalities ?

• A038336 → x ↦ x+1 → reverse 15 rows; no all-index equality certificate was found

Same fingerprint ?

• A038259 → basis A007318 → reverse
• A038271 → basis A013609 → x ↦ x+1
• A038301 → basis A013611 → reverse
• A038319 → D
• A038259 → basis A007318 → D
• A038220 → basis A013610 → x ↦ x/(x+1)
• A013618 → basis A013610 → basis A013610
• A038301 → basis A013611 → D
• A038301 → basis A013611
• A038336 → x ↦ x+1 → D
• A038259 → basis A007318
• A038336 → x ↦ x+1
• Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.
• A038319 → reverse

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.