image under θ+1 → x ↦ x+1

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Triangle begins:
  0:  1;
  1:  2,  2;
  2:  3,  6,   3;
  3:  4, 12,  12,   4;
  4:  5, 20,  30,  20,    5;
  5:  6, 30,  60,  60,   30,    6;
  6:  7, 42, 105, 140,  105,   42,   7;
  7:  8, 56, 168, 280,  280,  168,  56,   8;
  8:  9, 72, 252, 504,  630,  504, 252,  72,  9;
  9: 10, 90, 360, 840, 1260, 1260, 840, 360, 90, 10;

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

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n + 1 + (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; for(n = 1, N, my(i = n + 1); v[i] = ((n + 1 + (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 Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary), Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary), Laguerre’s theorem on differential operators
• Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary), Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by Geometric basis map (elementary), Laguerre’s theorem on differential operators
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from theta_plus_one of a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
    • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›

          • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
            • 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 basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
                • ↑ Parent-row provers of polytool (above)
                • ↑ Iterated real-rootedness preservers (above)
        • Parent-row provers of polytool polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Veronese sections of Pólya frequency sequences (above)
            • ↑ Parent-row provers of polytool (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from theta_plus_one of a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Laguerre’s theorem on multiplier sequences (above)
• Geometric basis map (elementary) polytool ? step ›

  • from derivative of basis-map coefficients of A scaled Hermite triangle.?
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from basis-map coefficients of A scaled Hermite triangle.?
        • 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 basis-map coefficients of A scaled Hermite triangle.?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from theta_plus_one of a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
    • ↑ Liu–Wang theorem for three-term recurrences (above)
    • ↑ Laguerre’s theorem on multiplier sequences (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from derivative of basis-map coefficients of A scaled Hermite triangle.?
    • ↑ Laguerre’s theorem on differential operators (above)
• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from shift_x_plus_one of basis-map coefficients of A scaled Hermite triangle.?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis-map coefficients of A scaled Hermite triangle.?
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from shift_x_plus_one of basis-map coefficients of A scaled Hermite triangle.?
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Geometric basis map (elementary) polytool ? step ›

          • from basis-map coefficients of A scaled Hermite triangle.?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A124860 → basis A013609
• A131431 → (θ+1)² → x ↦ x+1
• A003506 → x ↦ x+1 → basis A013610
• A134058 → basis A013609
• A038221 → basis A013609
• A003506 → D → basis A013609
• A094305 → D
• A085880 → D → D
• A156992 → D
• A027465 → basis A013611
• veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).
• A085881 → basis A007318 → basis A013610
• A134059 → D
• basis-map coefficients of A026729 → D → basis A007318
• A124860: A Jacobsthal-Pascal triangle.
• basis-map coefficients of A112227 → (θ+1)² → basis A007318
• A124860 → D
• Second weight array of Pascal's triangle (formatted as a rectangle), by antidiagonals.
• A178821: Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.
• A085880: Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A00010…
• basis-map coefficients of A026729, shifted → (θ+1)² → basis A007318
• A356546 → x ↦ x+1 → basis A013610
• Hadamard-factor parent of A127773 → x ↦ x+1
• A038330 → basis A013611 → basis A013609
• A038325 → D
• basis-map coefficients of A026729 → (θ+1)² → basis A007318
• A127648 → (θ+1)² → x ↦ x+1
• A135065 → x ↦ x+1 → basis A013610
• A356546: Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).
• A178821 → basis A013609
• A356546 → D
• A038208 → D → basis A013609
• basis-map coefficients of A112227 → θ+1 → x ↦ x+1
• A038208: Triangle whose (i,j)-th entry is binomial(i,j)*2^i.
• A178820 → basis A007318 → basis A013610
• A376827: T(n, k) = binomial(n, k)*hypergeom([(1 - n)/2, -n/2], [1], 4).
• A134309 → basis A007318
• A038221 → basis A013609 → D
• A135065 → basis A013609
• A135089 → basis A013609
• A094305 → x ↦ x+1 → basis A013610
• Riordan array (1-x,x/(1-x)).
• A196347 → basis A013609
• basis-map coefficients of A026729, shifted → basis A007318 → basis A013609
• A085881 → D
• A376827 → D
• image under θ+1 → (θ+1)² → x ↦ x+1
• basis-map coefficients of A113214, shifted → D → basis A013609
• basis-map coefficients of A113214, shifted → D → D
• A132440 → θ+1 → x ↦ x+1
• veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).
• A131431 → θ+1 → basis A007318
• A038330 → basis A013611 → D
• A178821 → D
• A178822 → D
• basis-map coefficients of Triangle read by rows: row n consists of n-1 zeros followed by Fibonacci(n).
• A085880 → basis A013609 → D
• A038325 → basis A007318 → basis A013610
• A038221 → D
• A131431 → x ↦ x+1 → basis A013609
• A131431 → x ↦ x+1 → D
• A097805: Number of compositions of n with k parts, T(n, k) = binomial(n-1, k-1) for n, k >= 1 and T(n, 0) = 0^n, triang…
• A028326 → basis A007318 → basis A013610
• A132200: Numbers in (4,4)-Pascal triangle .
• A258758: Triangle T(n,k) = C(n+k-1,k)*C(2*n-1,n-k).
• A134346: Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).
• A038221 → x ↦ x+1 → basis A013610
• A028326 → basis A013609
• basis-map coefficients of A113214, shifted → x ↦ x+1 → basis A013610
• A038208 → D → D
• A120910 → basis A013610 → D
• A038325 → basis A013609
• basis-map coefficients of A026729, shifted → basis A007318
• A219570: Triangular array read by rows. T(n,k) is the number of necklaces (turning over is not allowed) of n labeled b…
• A178820: Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.
• A196347 → basis A013609 → D
• A028326 → basis A013609 → basis A013609
• basis-map coefficients of A112227 → x ↦ x+1
• A134059: Triangle T(n, k) = 3*binomial(n,k) with T(0, 0) = 1, read by rows.
• A027465 → basis A013611 → D
• A134058: Triangle T(n, k) = 2*binomial(n, k) with T(0, 0) = 1, read by rows.
• A134346 → basis A013609
• A134346 → D
• A120910 → basis A007318 → basis A013611
• A007318 → ×(−(1/2)n+(1/2)n²) → D
• A356546 → basis A013609 → basis A013609
• A344565: Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).
• A003506 → D
• A038338 → x ↦ x+1 → basis A013610
• basis-map coefficients of A026729, shifted → θ+1 → x ↦ x+1
• basis-map coefficients of A026729 → basis A007318
• A135838 → basis A013609
• A135089: Triangle T(n,k) = 5*binomial(n,k) with T(0,0) = 1, read by rows.
• A135065 → D
• A258758 → D
• A129533: Array read by antidiagonals: T(n,k) = binomial(n+1,2)*binomial(n+k,n+1) for 0 <= k <= n.
• A038338 → D
• A156992 → basis A013609 → basis A013609
• A156992 → basis A013609 → D
• A178822 → basis A013609 → D
• basis-map coefficients of A026729, shifted → D → basis A007318
• A178821 → basis A013609 → D
• A178822 → x ↦ x+1 → basis A013610
• A094305: Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).
• A120910 → basis A013610
• A038338 → basis A013609
• A127773 → θ+1 → x ↦ x+1
• veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).
• A028326 → basis A013609 → D
• A196347 → D
• A196347: Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).
• A038338 → basis A013609 → D
• A135089 → D
• A135838: Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).
• A178822 → basis A013609 → basis A013609
• A129186 → basis A007318
• A178822 → basis A013609
• A178820 → basis A013609 → basis A013609
• A028326 → D
• A178820 → basis A013609 → D
• A038325 → basis A013609 → basis A013609
• A085880 → basis A013609
• A038221: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.
• A135065: A127733 * A007318 as infinite lower triangular matrices.
• A038208 → D
• A085880 → D
• basis-map coefficients of A030528, shifted → (θ+1)² → x ↦ x+1
• A027465 → basis A013611 → basis A013609
• A156992 → basis A013609
• A178822: Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.
• A136572 → (θ+1)² → basis A007318
• veronese parent of Binomial coefficients C(n,k) with n-k odd, read by rows.
• A120910 → basis A013610 → basis A013609
• A356546 → basis A013609 → D
• basis-map coefficients of A026729 → basis A007318 → basis A013609
• A376827 → basis A013609
• A127952: Triangle read by rows, T(n,k) = (n+1)*binomial(n-1,k-1).
• A380114: Triangle read by rows: The convolution triangle of 2^n, where the convolution triangle of a sequence is define…
• veronese parent of Binomial coefficients C(n,k) with n-k even, read by rows.
• A085880 → basis A013609 → basis A013609
• A085881: Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).
• A094305 → basis A013609
• basis-map coefficients of A113214, shifted → D
• A156992: Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.
• A132200 → D
• A038325 → basis A013609 → D
• A028326: Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).
• A344565 → basis A013609
• basis-map coefficients of A113214, shifted → basis A013609 → D
• A003506 → D → D
• A344565 → D
• A003506: Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1 <= k <= n.
• A156992 → basis A007318 → basis A013610
• A196347 → basis A013609 → basis A013609
• basis-map coefficients of A021012 → x ↦ x+1
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.
• A237765: Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.
• basis-map coefficients of A113214, shifted → basis A013609 → basis A013609
• A027465 → D → basis A013611
• A028326 → D → D
• A038279 → basis A013610 → basis A013611
• A038330 → basis A013611
• A085881 → basis A013609
• A085880 → D → basis A013609
• A131431 → x ↦ x+1
• basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0
• A134059 → basis A013609
• A007318: Binomial
• A178821 → basis A013609 → basis A013609
• A216973: Exponential Riordan array [x*exp(x),x].
• A135838 → D
• A178820 → D
• A178821 → basis A007318 → basis A013610
• A038338 → basis A013609 → basis A013609
• A127773 → (θ+1)² → x ↦ x+1
• A132200 → basis A013609
• basis-map coefficients of A113214, shifted → basis A013609
• A085881 → basis A013609 → D
• A130123 → θ+1 → basis A007318
• A356546 → basis A013609
• basis-map coefficients of A030528, shifted → θ+1 → basis A007318
• A178820 → basis A013609
• basis-map coefficients of A127647 → basis A013609
• A085880 → x ↦ x+1 → basis A013610
• basis-map coefficients of A127647 → D
• A038325: Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.
• A196347 → x ↦ x+1 → basis A013610
• A003506 → basis A013609
• basis-map coefficients of A112227 → x ↦ x+1 → basis A013609
• A258758 → basis A013609
• A085881 → basis A013609 → basis A013609
• basis-map coefficients of A021012 → θ+1 → basis A007318

References

1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
3. [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
4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
6. [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
7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.