A007318 → −D²+(−1+n)D+1+n → D+1

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
  0:   1;
  1:   4,    2;
  2:  11,   14,    3;
  3:  22,   54,   30,     4;
  4:  37,  140,  150,    52,     5;
  5:  56,  290,  480,   320,    80,     6;
  6:  79,  522, 1185,  1220,   585,   114,    7;
  7: 106,  854, 2478,  3500,  2590,   966,  154,    8;
  8: 137, 1304, 4620,  8344,  8470,  4872, 1484,  200,   9;
  9: 172, 1890, 7920, 17472, 22680, 17892, 8400, 2160, 252, 10;

The row polynomials satisfy (n^6-6*n^5+7*n^4+12*n^3-14*n^2-12*n)*P(n,x) = (n^6-3*n^5-5*n^4+15*n^3-14*n^2+6*n+36+(3*n^6-9*n^5-15*n^4+45*n^3-48*n^2+24*n+120)*x+(3*n^6-9*n^5-15*n^4+45*n^3-60*n^2+36*n+144)*x^2+(n^6-3*n^5-5*n^4+15*n^3-32*n^2+24*n+72)*x^3-(6*n^2-6*n-12)*x^4)*P(n-3,x) - (6*n^2-6*n-12+(24*n^2-24*n-48)*x+(36*n^2-36*n-72)*x^2+(24*n^2-24*n-48)*x^3+(6*n^2-6*n-12)*x^4)*d/dx P(n-3,x) - (3*n^6-12*n^5-9*n^4+60*n^3-18*n^2-12*n+24+(6*n^6-24*n^5-18*n^4+120*n^3-48*n^2+72)*x+(3*n^6-12*n^5-9*n^4+60*n^3-42*n^2+36*n+72)*x^2-(12*n^2-24*n-24)*x^3)*P(n-2,x) + (12*n^2-24*n-24+(36*n^2-72*n-72)*x+(36*n^2-72*n-72)*x^2+(12*n^2-24*n-24)*x^3)*d/dx P(n-2,x) + (3*n^6-15*n^5+3*n^4+63*n^3-36*n^2-54*n+(3*n^6-15*n^5+3*n^4+63*n^3-42*n^2-36*n)*x-(6*n^2-18*n)*x^2)*P(n-1,x) - (6*n^2-18*n+(12*n^2-36*n)*x+(6*n^2-18*n)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 4 + 2*x, P(2,x) = 11 + 14*x + 3*x^2, P(3,x) = 22 + 54*x + 30*x^2 + 4*x^3.

P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[2, x_] := 11 + 14*x + 3*x^2;
P[3, x_] := 22 + 54*x + 30*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((n^6 - 3*n^5 - 5*n^4 + 15*n^3 - 14*n^2 + 6*n + 36 + (3*n^6 - 9*n^5 - 15*n^4 + 45*n^3 - 48*n^2 + 24*n + 120)*x + (3*n^6 - 9*n^5 - 15*n^4 + 45*n^3 - 60*n^2 + 36*n + 144)*x^2 + (n^6 - 3*n^5 - 5*n^4 + 15*n^3 - 32*n^2 + 24*n + 72)*x^3 - (6*n^2 - 6*n - 12)*x^4)*P[n - 3, x] - (6*n^2 - 6*n - 12 + (24*n^2 - 24*n - 48)*x + (36*n^2 - 36*n - 72)*x^2 + (24*n^2 - 24*n - 48)*x^3 + (6*n^2 - 6*n - 12)*x^4)*D[P[n - 3, x], x] - (3*n^6 - 12*n^5 - 9*n^4 + 60*n^3 - 18*n^2 - 12*n + 24 + (6*n^6 - 24*n^5 - 18*n^4 + 120*n^3 - 48*n^2 + 72)*x + (3*n^6 - 12*n^5 - 9*n^4 + 60*n^3 - 42*n^2 + 36*n + 72)*x^2 - (12*n^2 - 24*n - 24)*x^3)*P[n - 2, x] + (12*n^2 - 24*n - 24 + (36*n^2 - 72*n - 72)*x + (36*n^2 - 72*n - 72)*x^2 + (12*n^2 - 24*n - 24)*x^3)*D[P[n - 2, x], x] + (3*n^6 - 15*n^5 + 3*n^4 + 63*n^3 - 36*n^2 - 54*n + (3*n^6 - 15*n^5 + 3*n^4 + 63*n^3 - 42*n^2 - 36*n)*x - (6*n^2 - 18*n)*x^2)*P[n - 1, x] - (6*n^2 - 18*n + (12*n^2 - 36*n)*x + (6*n^2 - 18*n)*x^2)*D[P[n - 1, x], x])/(n^6 - 6*n^5 + 7*n^4 + 12*n^3 - 14*n^2 - 12*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 2*x; v[3] = 11 + 14*x + 3*x^2; v[4] = 22 + 54*x + 30*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((n^6 - 3*n^5 - 5*n^4 + 15*n^3 - 14*n^2 + 6*n + 36 + (3*n^6 - 9*n^5 - 15*n^4 + 45*n^3 - 48*n^2 + 24*n + 120)*x + (3*n^6 - 9*n^5 - 15*n^4 + 45*n^3 - 60*n^2 + 36*n + 144)*x^2 + (n^6 - 3*n^5 - 5*n^4 + 15*n^3 - 32*n^2 + 24*n + 72)*x^3 - (6*n^2 - 6*n - 12)*x^4)*v[i - 3] - (6*n^2 - 6*n - 12 + (24*n^2 - 24*n - 48)*x + (36*n^2 - 36*n - 72)*x^2 + (24*n^2 - 24*n - 48)*x^3 + (6*n^2 - 6*n - 12)*x^4)*deriv(v[i - 3], x) - (3*n^6 - 12*n^5 - 9*n^4 + 60*n^3 - 18*n^2 - 12*n + 24 + (6*n^6 - 24*n^5 - 18*n^4 + 120*n^3 - 48*n^2 + 72)*x + (3*n^6 - 12*n^5 - 9*n^4 + 60*n^3 - 42*n^2 + 36*n + 72)*x^2 - (12*n^2 - 24*n - 24)*x^3)*v[i - 2] + (12*n^2 - 24*n - 24 + (36*n^2 - 72*n - 72)*x + (36*n^2 - 72*n - 72)*x^2 + (12*n^2 - 24*n - 24)*x^3)*deriv(v[i - 2], x) + (3*n^6 - 15*n^5 + 3*n^4 + 63*n^3 - 36*n^2 - 54*n + (3*n^6 - 15*n^5 + 3*n^4 + 63*n^3 - 42*n^2 - 36*n)*x - (6*n^2 - 18*n)*x^2)*v[i - 1] - (6*n^2 - 18*n + (12*n^2 - 36*n)*x + (6*n^2 - 18*n)*x^2)*deriv(v[i - 1], x))/(n^6 - 6*n^5 + 7*n^4 + 12*n^3 - 14*n^2 - 12*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)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-ac1cdf02143513b6
• Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-ac1cdf02143513b6
• Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-ac1cdf02143513b6

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• preserver_image derivative_polynomial, shift 0 e-ac1cdf02143513b6

  Laguerre’s theorem on differential operators [SP14] polytool ?

  • premise (sequence): Triangle read by rows: T(n, k) = (n-k+1)*(k+1)*binomial(n, k).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image euler_polynomial, shift 0 e-fe7c4fb58c0b00e3

      Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

      • premise (sequence): Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • preserver_image derivative_polynomial, shift 0 e-5c65a1a70531c253

          Laguerre’s theorem on differential operators [SP14] polytool ?

          • premise (sequence): binomial
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • theorem_check liu_wang e-317eb4ccca60c6ac root: theorem check

              Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

            • preserver_image reversal, shift 0 e-033cf842aad3bdf7

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): binomial
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ theorem_check liu_wang e-317eb4ccca60c6ac (expanded above)
                • basis_map A007318 e-095baa2a9dfe188f

                  Geometric basis map (elementary) polytool ?

                  • premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
                    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                    • veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d

                      Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ?

                      • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
                        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                        • parent_provers periodic_product e-1991a148b9538b93 root: theorem check

                          Parent-row provers of polytool polytool ?

                        • preserver_image mobius, shift 0 e-12303ad0e0b9f8eb

                          Affine, reversal and Möbius substitutions (elementary) polytool ?

                          • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
                            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                            • ↵ parent_provers periodic_product e-1991a148b9538b93 (expanded above)
                    • parent_provers periodic_product e-b7eca039219a7f53 root: theorem check

                      Parent-row provers of polytool polytool ?

                    • preserver_image mobius, shift 0 e-7e44b66e04c497d6

                      Affine, reversal and Möbius substitutions (elementary) polytool ?

                      • premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
                        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                        • ↵ veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d (expanded above)
                        • ↵ parent_provers periodic_product e-b7eca039219a7f53 (expanded above)
                • parent_provers periodic_product e-13983862a411779c root: theorem check

                  Parent-row provers of polytool polytool ?

                • preserver_image reversal, shift 0 e-989c95e5c893fce9

                  Affine, reversal and Möbius substitutions (elementary) polytool ?

                  • premise (sequence): Riordan array (1-x,x/(1-x)).
                    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                    • parent_provers periodic_product e-b62074ba45057e82 root: theorem check

                      Parent-row provers of polytool polytool ?

                • gamma_polynomial scale 1, shift 0 e-c76adc2a99cd3dea

                  Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

                  • premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
                    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                    • preserver_image reversal, shift 0 e-0e058c428dfdfeb6

                      Affine, reversal and Möbius substitutions (elementary) polytool ?

                      • premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
                        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                        • parent_provers periodic_product e-d82c9b48e4b31a6b root: theorem check

                          Parent-row provers of polytool polytool ?

                    • ↵ parent_provers periodic_product e-d82c9b48e4b31a6b (expanded above)
                • preserver_image affine, shift 0 e-3a52de7742ed9314

                  Affine, reversal and Möbius substitutions (elementary) polytool ?

                  • premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
                    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                    • ↵ veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d (expanded above)
                    • ↵ parent_provers periodic_product e-b7eca039219a7f53 (expanded above)
                    • ↵ preserver_image mobius, shift 0 e-7e44b66e04c497d6 (expanded above)
            • ↵ basis_map A007318 e-095baa2a9dfe188f (expanded above)
            • ↵ parent_provers periodic_product e-13983862a411779c (expanded above)
            • ↵ preserver_image reversal, shift 0 e-989c95e5c893fce9 (expanded above)
            • ↵ gamma_polynomial scale 1, shift 0 e-c76adc2a99cd3dea (expanded above)
            • ↵ preserver_image affine, shift 0 e-3a52de7742ed9314 (expanded above)
        • preserver_image reversal, shift 0 e-c2af0ae48d7f11c8

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix with 2's o…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • preserver_image euler_polynomial, shift 0 e-a112169ce63438eb

              Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

              • premise (sequence): binomial
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ theorem_check liu_wang e-317eb4ccca60c6ac (expanded above)
                • ↵ preserver_image reversal, shift 0 e-033cf842aad3bdf7 (expanded above)
                • ↵ basis_map A007318 e-095baa2a9dfe188f (expanded above)
                • ↵ parent_provers periodic_product e-13983862a411779c (expanded above)
                • ↵ preserver_image reversal, shift 0 e-989c95e5c893fce9 (expanded above)
                • ↵ gamma_polynomial scale 1, shift 0 e-c76adc2a99cd3dea (expanded above)
                • ↵ preserver_image affine, shift 0 e-3a52de7742ed9314 (expanded above)
            • preserver_image reversal, shift 0 e-ac2f606fc4a9c0c0

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ preserver_image derivative_polynomial, shift 0 e-5c65a1a70531c253 (expanded above)
                • preserver_image derivative_polynomial, shift 0 e-d2751b1a8ff0ac45

                  Laguerre’s theorem on differential operators [SP14] polytool ?

                  • premise (sequence): binomial
                    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                    • ↵ theorem_check liu_wang e-317eb4ccca60c6ac (expanded above)
                    • ↵ preserver_image reversal, shift 0 e-033cf842aad3bdf7 (expanded above)
                    • ↵ basis_map A007318 e-095baa2a9dfe188f (expanded above)
                    • ↵ parent_provers periodic_product e-13983862a411779c (expanded above)
                    • ↵ preserver_image reversal, shift 0 e-989c95e5c893fce9 (expanded above)
                    • ↵ gamma_polynomial scale 1, shift 0 e-c76adc2a99cd3dea (expanded above)
                    • ↵ preserver_image affine, shift 0 e-3a52de7742ed9314 (expanded above)
            • preserver_image euler_polynomial, shift 0 e-becc9d9cf02b45c7

              Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

              • premise (sequence): binomial
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ theorem_check liu_wang e-317eb4ccca60c6ac (expanded above)
                • ↵ preserver_image reversal, shift 0 e-033cf842aad3bdf7 (expanded above)
                • ↵ basis_map A007318 e-095baa2a9dfe188f (expanded above)
                • ↵ parent_provers periodic_product e-13983862a411779c (expanded above)
                • ↵ preserver_image reversal, shift 0 e-989c95e5c893fce9 (expanded above)
                • ↵ gamma_polynomial scale 1, shift 0 e-c76adc2a99cd3dea (expanded above)
                • ↵ preserver_image affine, shift 0 e-3a52de7742ed9314 (expanded above)
        • ↵ preserver_image derivative_polynomial, shift 0 e-d2751b1a8ff0ac45 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
4. [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
5. [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
6. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.