A376827 → basis A007318

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Submission text ?

Triangle begins:
  0:       1;
  1:       2,       1;
  2:      12,      12,        3;
  3:      56,      84,       42,        7;
  4:     304,     608,      456,      152,       19;
  5:    1632,    4080,     4080,     2040,      510,      51;
  6:    9024,   27072,    33840,    22560,     8460,    1692,     141;
  7:   50304,  176064,   264096,   220080,   110040,   33012,    5502,    393;
  8:  283392, 1133568,  1983744,  1983744,  1239840,  495936,  123984,  17712,  1107;
  9: 1607168, 7232256, 14464512, 16875264, 12656448, 6328224, 2109408, 452016, 56502, 3139;

The row polynomials satisfy n*P(n,x) = (12*n-12+(12*n-12)*x+(3*n-3)*x^2)*P(n-2,x) + (4*n-2+(2*n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((12*n - 12 + (12*n - 12)*x + (3*n - 3)*x^2)*v[i - 2] + (4*n - 2 + (2*n - 1)*x)*v[i - 1])/(n)); 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]
  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 Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from T(n, k) = binomial(n, k)*hypergeom([(1 - n)/2, -n/2], [1], 4).?
    • 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
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from T(n, k) = binomial(n, k)*hypergeom([(1 - n)/2, -n/2], [1], 4).?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from T(n, k) = binomial(n, k)*hypergeom([(1 - n)/2, -n/2], [1], 4).?
    • ↑ Row-hypergeometric rows (multiplier sequences) (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A003506 → x ↦ x+1
• A356546 → basis A013611
• A038338 → basis A013611
• A131431 → D → basis A013609
• A344565 → basis A013611
• A038338 → x ↦ x+1
• A131431 → x ↦ x+1 → basis A007318
• A085881 → basis A007318 → D
• A356546 → x ↦ x+1
• A196347 → x ↦ x+1
• A120909 → basis A013610
• basis-map coefficients of A112227 → (θ+1)² → basis A013609
• A003506 → D → x ↦ x+1
• A135065 → x ↦ x+1
• A356546 → x ↦ x/(x+1)
• A027466 → basis A013611
• A134309 → basis A013609
• A085880 → x ↦ x+1
• A038325 → x ↦ x/(x+1)
• A027465 → D → basis A013609
• A120910: Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-…
• basis-map coefficients of A026729 → basis A007318 → basis A013611
• image under θ+1 → (θ+1)² → basis A013609
• image under θ+1 → basis A013609
• A196347 → x ↦ x+1 → D
• A085881 → basis A013611
• A003506 → D → basis A013611
• basis-map coefficients of A026729 → basis A013609 → basis A013610
• A038291 → basis A013609 → basis A013610
• A131431 → x ↦ x+1 → basis A013611
• A135065 → x ↦ x+1 → D
• A131436 → basis A013609
• A120909 → basis A013610 → basis A013610
• A156992 → basis A013609 → basis A007318
• A094305 → x ↦ x+1 → D
• A038338 → x ↦ x+1 → reverse
• A085881 → basis A007318 → reverse
• A136572 → basis A013609 → D
• A178822 → basis A013609 → x ↦ x+1
• A038208 → basis A013611
• A013609: Triangle of coefficients in expansion of (1+2*x)^n.
• A156992 → basis A013611
• A344565 → x ↦ x/(x+1)
• A013609 → basis A013610 → D
• basis-map coefficients of A026729 → basis A013609
• basis-map coefficients of A026729, shifted → basis A013609 → D
• A134058 → basis A013611
• A038208 → D → basis A013611
• A038325 → basis A013609 → basis A007318
• A013609 → D → basis A013610
• basis-map coefficients of A127647 → basis A013611
• A356546 → basis A013611 → D
• A028326 → basis A013611 → basis A013610
• A013609 → D → D
• A085881 → basis A013611 → D
• A038208 → basis A007318 → D
• A178821 → basis A007318 → D
• A135089 → basis A013611
• basis-map coefficients of A112227 → θ+1 → basis A013609
• A085880 → D → basis A013611
• A356546 → basis A013611 → basis A013610
• basis-map coefficients of A112227 → basis A013609 → basis A013610
• A136572 → (θ+1)² → basis A013609
• A038330 → basis A013611 → x ↦ x/(x+1)
• A178822 → x ↦ x+1 → reverse
• A134059 → basis A013611
• basis-map coefficients of A026729 → (θ+1)² → basis A013609
• A028326 → basis A007318
• A120910 → basis A007318 → basis A013609
• A134058 → basis A007318
• A136572 → θ+1 → basis A013609
• basis-map coefficients of A026729, shifted → basis A007318 → x ↦ x+1
• A120910 → basis A013610 → basis A013611
• A124860 → x ↦ x+1
• A028326 → basis A013611 → D
• Veronese parent of A098172 → x ↦ x/(x+1)
• A130123 → θ+1 → basis A013609
• A156992 → basis A007318 → D
• A120910 → D
• A038221 → x ↦ x+1
• A156992 → basis A007318
• A038330 → basis A013609
• A135838 → x ↦ x/(x+1)
• A028326 → basis A013609 → basis A013611
• A178821 → basis A013611 → reverse
• basis-map coefficients of A112227 → basis A013609
• A156992 → x ↦ x/(x+1)
• A038338 → basis A013609 → basis A007318
• A028326 → x ↦ x/(x+1)
• A131431 → basis A013609
• A085880 → x ↦ x+1 → reverse
• A178821 → x ↦ x/(x+1)
• A178822 → x ↦ x+1
• A178821 → basis A013611
• A258758 → basis A013611
• A124860 → basis A013611
• A178820 → basis A013611 → reverse
• A085880 → basis A013609 → basis A013611
• A178822 → basis A013611 → basis A013610
• A178821 → basis A013611 → D
• A178821 → basis A007318 → reverse
• A130123 → D → basis A013609
• A178822 → x ↦ x/(x+1)
• A135089 → basis A007318
• basis-map coefficients of A113214, shifted → x ↦ x+1
• A038329 → x ↦ x+1 → basis A013610
• A134346 → x ↦ x+1
• A120909 → basis A013610 → reverse
• A127648 → D → basis A013609
• A120909 → basis A013610 → D
• A094305 → x ↦ x+1
• A027466 → basis A013611 → D
• A132200 → basis A007318
• A038338 → x ↦ x/(x+1)
• A038221 → x ↦ x+1 → D
• A085881 → x ↦ x/(x+1)
• A159854 → x ↦ x/(x+1)
• basis-map coefficients of A113214, shifted → basis A013611
• basis-map coefficients of A021012 → basis A013609
• A135065 → x ↦ x/(x+1)
• A131431 → θ+1 → basis A013609
• A085880 → basis A013611
• A013609 → D
• A085881 → basis A013611 → basis A013610
• A178821 → basis A013611 → basis A013610
• A028326 → basis A007318 → D
• A156992 → basis A013611 → D
• A038221 → basis A013611 → D
• A038325 → basis A013611
• A013609 → basis A013610
• A013609 → basis A013610 → basis A013610
• basis-map coefficients of A026729, shifted → D → basis A013609
• A028326 → basis A013609 → x ↦ x+1
• A038208 → x ↦ x/(x+1)
• basis-map coefficients of A113214, shifted → D → basis A007318
• A085880 → basis A013611 → D
• A178822 → x ↦ x+1 → D
• A038330 → basis A013609 → reverse
• A085880 → D → basis A007318
• A136572 → D → basis A013609
• basis-map coefficients of A127647 → basis A007318
• A094305 → x ↦ x/(x+1)
• A028326 → basis A013611
• A038221 → basis A013611
• A178820 → basis A007318 → reverse
• A178820 → basis A007318 → D
• A136521 → basis A013609
• A085880 → basis A013611 → basis A013610
• A178821 → basis A013609 → basis A013611
• A178822 → basis A013611
• A135065 → basis A013611
• A038330 → basis A013611 → basis A007318
• A038325 → basis A007318 → reverse
• basis-map coefficients of A026729, shifted → θ+1 → basis A013609
• A132200 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013609 → D
• Veronese parent of A098172 → x ↦ x/(x+1)
• A134346 → x ↦ x/(x+1)
• A356546 → basis A013611 → reverse
• A134059 → basis A007318
• A132440 → θ+1 → basis A013609
• basis-map coefficients of A030528, shifted → θ+1 → basis A013609
• A135065 → x ↦ x+1 → reverse
• A085881 → basis A007318
• A038208 → basis A007318
• A136572 → basis A013609
• A085880 → x ↦ x/(x+1)
• A085880 → basis A013609 → x ↦ x+1
• A085881 → basis A013611 → reverse
• basis-map coefficients of A026729 → basis A013609 → reverse
• basis-map coefficients of A112227 → x ↦ x+1 → basis A007318
• A038208 → basis A013611 → D
• A356546 → basis A013609 → basis A013611
• A135838 → x ↦ x+1
• basis-map coefficients of A113214, shifted → x ↦ x/(x+1)
• A038330 → basis A013609 → D
• A178820 → x ↦ x/(x+1)
• A038325 → basis A007318 → D
• A147716 → basis A013610 → basis A013610
• basis-map coefficients of A112227 → x ↦ x+1 → basis A013611
• A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
• basis-map coefficients of A030528, shifted → basis A013609
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x+1
• basis-map coefficients of A021012 → θ+1 → basis A013609
• basis-map coefficients of A026729, shifted → basis A007318 → basis A013611
• basis-map coefficients of A026729, shifted → basis A013609 → basis A013610
• A136572 → basis A013609 → basis A013610
• A178822 → basis A013609 → basis A013611
• basis-map coefficients of A026729 → basis A013609 → D
• A131431 → x ↦ x+1 → x ↦ x/(x+1)
• basis-map coefficients of A030528, shifted → D → basis A013609
• A130123 → basis A013609
• basis-map coefficients of A026729, shifted → basis A013609
• A178820 → basis A013611 → basis A013610
• basis-map coefficients of A026729 → D → basis A013609
• A356546 → x ↦ x+1 → reverse
• A135838 → basis A013611
• A003506 → x ↦ x/(x+1)
• A196347 → basis A013611 → D
• A376827 → basis A013611
• A094305 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013609 → basis A013610
• A178820 → basis A013609 → basis A013611
• A120909: Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maxi…
• A130123 → basis A013609 → D
• basis-map coefficients of A113214, shifted → D → basis A013611
• A094305 → x ↦ x+1 → reverse
• basis-map coefficients of A113214, shifted → basis A013609 → x ↦ x+1
• A156992 → basis A013611 → basis A013610
• basis-map coefficients of A026729, shifted → (θ+1)² → basis A013609
• A216973 → basis A013611
• A196347 → basis A013611
• A178820 → basis A013611
• basis-map coefficients of A112227 → basis A013609 → D
• basis-map coefficients of A113214, shifted → basis A013609 → basis A013611
• A178822 → basis A013611 → D
• A344565 → basis A007318
• Veronese parent of A098172 → x ↦ x/(x+1)
• A120909 → D
• A178820 → basis A013611 → D
• A038221 → basis A013611 → basis A013610
• basis-map coefficients of A127647 → x ↦ x/(x+1)
• A196347 → basis A013609 → basis A007318
• A038338 → x ↦ x+1 → D
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x/(x+1)
• A028326 → basis A007318 → reverse
• A134346 → basis A013611
• A027465 → basis A013611 → basis A013611
• A132440 → basis A013609
• A356546 → x ↦ x+1 → D
• A258758 → basis A007318
• basis-map coefficients of A030528, shifted → (θ+1)² → basis A013609
• A178821 → basis A007318
• A038325 → basis A007318
• A376827 → x ↦ x/(x+1)
• A129186 → basis A013609
• A038221 → basis A013609 → basis A013611
• A178820 → basis A007318
• image under θ+1 → θ+1 → D → basis A013609
• A124860 → x ↦ x/(x+1)
• A196347 → basis A013611 → basis A013610
• A120910 → D → D

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.