A038338 → (θ+1)²

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Submission text ?

Triangle begins:
  0:          144,           1152,            1296;
  1:         1728,          20736,           46656,            27648;
  2:        20736,         331776,         1119744,          1327104,           518400;
  3:       248832,        4976640,        22394880,         39813120,         31104000,           8957952;
  4:      2985984,       71663616,       403107840,        955514880,       1119744000,         644972544,         146313216;
  5:     35831808,     1003290624,      6772211712,      20065812480,      31352832000,       27088846848,       12290310144,        2293235712;
  6:    429981696,    13759414272,    108355387392,     385263599616,     752467968000,      866843099136,      589934886912,      220150628352,      34828517376;
  7:   5159780352,   185752092672,   1671768834048,    6934744793088,   16253308108800,    23404763676672,    21237655928832,    11888133931008,    3761479876608,     515978035200;
  8:  61917364224,  2476694568960,  25076532510720,  118881339310080,  325066162176000,   561714328240128,   637129677864960,   475525357240320,  225688792596480,   61917364224000,   7492001071104;
  9: 743008370688, 32692368310272, 367789143490560, 1961542098616320, 6129819058176000, 12357715221282816, 16820223495634944, 15692336788930560, 9930306874245120, 4086546038784000, 988944141385728, 106993205379072;

The row polynomials satisfy (1+(3*n+5)*x+(n^2+4*n+4)*x^2)*P(n,x) = (12+(36*n+108)*x+(12*n^2+108*n+204)*x^2+(12*n^2+72*n+108)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 1152*x + 1296*x^2.

P[0, x_] := 144 + 1152*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((12 + (36*n + 108)*x + (12*n^2 + 108*n + 204)*x^2 + (12*n^2 + 72*n + 108)*x^3)*P[n - 1, x])/(1 + (3*n + 5)*x + (n^2 + 4*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 144 + 1152*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((12 + (36*n + 108)*x + (12*n^2 + 108*n + 204)*x^2 + (12*n^2 + 72*n + 108)*x^3)*v[i - 1])/(1 + (3*n + 5)*x + (n^2 + 4*n + 4)*x^2)); 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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^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
    • 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)*12^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A003506 → (θ+1)²
• A003506 → D → (θ+1)²
• A038330 → basis A013611 → (θ+1)²
• basis-map coefficients of A026729 → basis A007318 → (θ+1)²
• A131431 → x ↦ x+1 → (θ+1)²
• A344565 → (θ+1)²
• basis-map coefficients of A113214, shifted → D → (θ+1)²
• basis-map coefficients of A113214, shifted → (θ+1)²
• A156992 → basis A013609 → (θ+1)²
• A135065 → (θ+1)²
• A028326 → D → (θ+1)²
• A120910 → basis A013610 → (θ+1)²
• A178821 → basis A013609 → (θ+1)²
• A376827 → (θ+1)²
• A028326 → basis A013609 → (θ+1)²
• A085880 → basis A013609 → (θ+1)²
• A196347 → (θ+1)²
• A178821 → (θ+1)²
• A356546 → (θ+1)²
• A125092 → reverse → D
• A125092 → reverse
• A178820 → basis A013609 → (θ+1)²
• A356546 → basis A013609 → (θ+1)²
• A178822 → (θ+1)²
• A134346 → (θ+1)²
• A038208 → (θ+1)²
• A038325 → (θ+1)²
• A038325 → basis A013609 → (θ+1)²
• A085881 → basis A013609 → (θ+1)²
• A038221 → (θ+1)²
• A085881 → (θ+1)²
• A085880 → (θ+1)²
• A124860 → (θ+1)²
• basis-map coefficients of A112227 → x ↦ x+1 → (θ+1)²
• basis-map coefficients of A113214, shifted → basis A013609 → (θ+1)²
• basis-map coefficients of A026729, shifted → basis A007318 → (θ+1)²
• A125092: Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).
• A196347 → basis A013609 → (θ+1)²
• A178822 → basis A013609 → (θ+1)²
• A156992 → (θ+1)²
• A027465 → basis A013611 → (θ+1)²
• basis-map coefficients of A127647 → (θ+1)²
• A178820 → (θ+1)²
• A135838 → (θ+1)²
• A028326 → (θ+1)²
• A038338 → basis A013609 → (θ+1)²
• A038221 → basis A013609 → (θ+1)²
• A085880 → D → (θ+1)²
• A038208 → D → (θ+1)²

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.