{"id":"n-52badbe8a6288624","revision":0,"first_index":0,"fingerprint":"f6-729ce679a0f8b563","name":"derivative of veronese_section_2_0 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["5"],["13","2"],["25","14"],["41","52","3"],["61","140","27"],["85","310","129","4"],["113","602","441","44"],["145","1064","1218","256","5"],["181","1752","2898","1056","65"],["221","2730","6174","3480","445","6"],["265","4070","12078","9768","2145","90"],["313","5852","22077","24288","8195","708","7"],["365","8164","38181","54912","26455","3900","119"],["421","11102","63063","114972","75075","16926","1057","8"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","5","15","39","96","228","528","1200","2688","5952","13056","28416","61440","132096","282624"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","6","0"],["8","-2","3"],["-16","-12","4"],["16","-8","1"]]},"terms":[{"coeff":{"coeffs":[["0","38","12","-2"],["8","-74","31","-1"],["16","-52","4","2"],["16","8","-7","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-16","4","16","-4"],["0","32","-64","38","-6"],["0","32","-48","18","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173160,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-ca6664009f4960b0"]},{"id":173161,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-ca6664009f4960b0"]},{"id":173162,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-ca6664009f4960b0"]}],"region_exact":{},"incoming_edges":[{"id":"e-ca6664009f4960b0","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-52badbe8a6288624","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":"1cd776b06b326237dc012564d9b39f2feed30b86bb403f8ef3507a6aa413f4f1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-cbf81d8f5ae9d341","role":"sequence","name":"veronese_section_2_0 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:   1;\n  1:   5;\n  2:  13,    2;\n  3:  25,   14;\n  4:  41,   52,    3;\n  5:  61,  140,   27;\n  6:  85,  310,  129,    4;\n  7: 113,  602,  441,   44;\n  8: 145, 1064, 1218,  256,  5;\n  9: 181, 1752, 2898, 1056, 65;\n","recurrence_sentence":"The row polynomials satisfy (16*n^3-16*n^2+8*n-(8*n^3+12*n^2+2*n-6)*x+(n^3+4*n^2+3*n)*x^2)*P(n,x) = (16*n^3+16*n^2+8*n+(8*n^3-52*n^2-74*n+38)*x-(7*n^3-4*n^2-31*n-12)*x^2+(n^3+2*n^2-n-2)*x^3)*P(n-1,x) + ((32*n^2+32*n-16)*x-(48*n^2+64*n-4)*x^2+(18*n^2+38*n+16)*x^3-(2*n^2+6*n+4)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n^3 + 16*n^2 + 8*n + (8*n^3 - 52*n^2 - 74*n + 38)*x - (7*n^3 - 4*n^2 - 31*n - 12)*x^2 + (n^3 + 2*n^2 - n - 2)*x^3)*P[n - 1, x] + ((32*n^2 + 32*n - 16)*x - (48*n^2 + 64*n - 4)*x^2 + (18*n^2 + 38*n + 16)*x^3 - (2*n^2 + 6*n + 4)*x^4)*D[P[n - 1, x], x])/(16*n^3 - 16*n^2 + 8*n - (8*n^3 + 12*n^2 + 2*n - 6)*x + (n^3 + 4*n^2 + 3*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((16*n^3 + 16*n^2 + 8*n + (8*n^3 - 52*n^2 - 74*n + 38)*x - (7*n^3 - 4*n^2 - 31*n - 12)*x^2 + (n^3 + 2*n^2 - n - 2)*x^3)*v[i - 1] + ((32*n^2 + 32*n - 16)*x - (48*n^2 + 64*n - 4)*x^2 + (18*n^2 + 38*n + 16)*x^3 - (2*n^2 + 6*n + 4)*x^4)*deriv(v[i - 1], x))/(16*n^3 - 16*n^2 + 8*n - (8*n^3 + 12*n^2 + 2*n - 6)*x + (n^3 + 4*n^2 + 3*n)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f48bed27dbc29c5f","steps":["Veronese 2","D"],"edges":["e-9f9d452726caee29","e-ca6664009f4960b0"],"text":"A391533 \u2192 Veronese 2 \u2192 D"},"headline":"A391533 \u2192 Veronese 2 \u2192 D","chain":null}