A376827 → x ↦ x/(x+1)

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Submission text ?

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

The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = (3*n-3+(18*n-18)*x+(39*n-39)*x^2+(36*n-36)*x^3+(12*n-12)*x^4)*P(n-2,x) + (2*n-1+(8*n-4)*x+(10*n-5)*x^2+(4*n-2)*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 3 + (18*n - 18)*x + (39*n - 39)*x^2 + (36*n - 36)*x^3 + (12*n - 12)*x^4)*P[n - 2, x] + (2*n - 1 + (8*n - 4)*x + (10*n - 5)*x^2 + (4*n - 2)*x^3)*P[n - 1, x])/(n + 2*n*x + n*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((3*n - 3 + (18*n - 18)*x + (39*n - 39)*x^2 + (36*n - 36)*x^3 + (12*n - 12)*x^4)*v[i - 2] + (2*n - 1 + (8*n - 4)*x + (10*n - 5)*x^2 + (4*n - 2)*x^3)*v[i - 1])/(n + 2*n*x + n*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 [-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 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)

Related ?

Same fingerprint ?

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

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.