{"id":"n-832225da1d62d7cf","revision":0,"first_index":1,"fingerprint":"f6-bfa3eeae29a25dff","name":"one_plus_derivative of shift_x_plus_one of T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["5","2"],["13","6"],["43","36","6"],["131","140","30"],["411","570","210","20"],["1275","2142","1050","140"],["3963","7896","5040","1120","70"],["12283","28296","22176","6720","630"],["38023","99630","93240","36960","5670","252"],["117503","345290","376200","184800","39270","2772"],["362605","1181796","1472130","868560","242550","27720","924"],["1117429","4001868","5615610","3878160","1351350","216216","12012"],["3439229","13429598","20978958","16636620","6996990","1477476","132132","3432"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","7","19","85","301","1211","4607","18089","70105","273775","1065835","4156285","16192645","63094435"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["4","7","-4"],["-3","-11","4"],["-2","4","0"],["1","0","0"]]},"terms":[{"coeff":{"coeffs":[["32","55","-12","-32"],["-13","-59","-24","16"],["-12","8","16","0"],["5","4","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["6","8","-30","-64","-32"],["-6","-38","-64","-32","0"],["-6","-14","-8","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174077,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-209d206005d79553"]},{"id":174078,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-209d206005d79553"]},{"id":174079,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-209d206005d79553"]}],"region_exact":{},"incoming_edges":[{"id":"e-209d206005d79553","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-832225da1d62d7cf","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"762d381157e50b0affed89950fc6f8211583c896448d0e4b0e6ac74f2a7079d5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-123dac0e8ec7cfb1","role":"sequence","name":"shift_x_plus_one of T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-bff2e4123dda8c59","kind":"fingerprint_related","node_a_id":"n-832225da1d62d7cf","node_b_id":"n-389e361771f4b4ac","fingerprint":"f6-bfa3eeae29a25dff","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-bfa3eeae29a25dff"},"other_id":"n-389e361771f4b4ac","other":{"id":"n-389e361771f4b4ac","name":"one_plus_derivative of shift_x_plus_one of Triangle read by rows: T(n,k) is k-th entry of the gamma-vector of the n-dimensional cyclohedron, 0 <= k <= floor(n/2).","first_index":0,"fingerprint":"f6-bfa3eeae29a25dff","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["5","2"]],"card_rows":[["1"],["1"],["5","2"],["13","6"]],"row_sums_preview":["1","1","7","19","85","301","1211","4607"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8bfdd58279be42d7","steps":["x \u21a6 x+1","D+1"],"edges":["e-0f1b9062159c7e91","e-5a84e90687e3aa52"],"text":"A391403 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A391403 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:     1;\n   2:     1;\n   3:     5,     2;\n   4:    13,     6;\n   5:    43,    36,     6;\n   6:   131,   140,    30;\n   7:   411,   570,   210,   20;\n   8:  1275,  2142,  1050,  140;\n   9:  3963,  7896,  5040, 1120,  70;\n  10: 12283, 28296, 22176, 6720, 630;\n","recurrence_sentence":"The row polynomials satisfy (n^3-2*n^2-3*n+4+(4*n^2-11*n+7)*x+(4*n-4)*x^2)*P(n,x) = (5*n^3-12*n^2-13*n+32+(4*n^3+8*n^2-59*n+55)*x+(16*n^2-24*n-12)*x^2+(16*n-32)*x^3)*P(n-1,x) - (6*n^2+6*n-6+(14*n^2+38*n-8)*x+(8*n^2+64*n+30)*x^2+(32*n+64)*x^3+32*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((5*n^3 - 12*n^2 - 13*n + 32 + (4*n^3 + 8*n^2 - 59*n + 55)*x + (16*n^2 - 24*n - 12)*x^2 + (16*n - 32)*x^3)*P[n - 1, x] - (6*n^2 + 6*n - 6 + (14*n^2 + 38*n - 8)*x + (8*n^2 + 64*n + 30)*x^2 + (32*n + 64)*x^3 + 32*x^4)*D[P[n - 1, x], x])/(n^3 - 2*n^2 - 3*n + 4 + (4*n^2 - 11*n + 7)*x + (4*n - 4)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((5*n^3 - 12*n^2 - 13*n + 32 + (4*n^3 + 8*n^2 - 59*n + 55)*x + (16*n^2 - 24*n - 12)*x^2 + (16*n - 32)*x^3)*v[i - 1] - (6*n^2 + 6*n - 6 + (14*n^2 + 38*n - 8)*x + (8*n^2 + 64*n + 30)*x^2 + (32*n + 64)*x^3 + 32*x^4)*deriv(v[i - 1], x))/(n^3 - 2*n^2 - 3*n + 4 + (4*n^2 - 11*n + 7)*x + (4*n - 4)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-4b168caf0bc63f8b","steps":["x \u21a6 x+1","D+1"],"edges":["e-502ba5da9494e9e3","e-209d206005d79553"],"text":"A089627 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A089627 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}