{"id":"n-ae7fa795b878b77c","revision":0,"first_index":0,"fingerprint":"f6-51638b1898cb1bac","name":"derivative of shift_x_plus_one of Coefficient triangle of certain polynomials N(4; m,x).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["5"],["42","30"],["252","336","105"],["1320","2520","1512","280"],["6435","15840","13860","5040","630"],["30030","90090","102960","55440","13860","1260"],["136136","480480","675675","480480","180180","33264","2310"],["604656","2450448","4084080","3603600","1801800","504504","72072","3960"],["2645370","12093120","23279256","24504480","15315300","5765760","1261260","144144","6435"],["11440660","58198140","126977760","155195040","116396280","55135080","16336320","2882880","270270","10010"],["49031400","274575840","669278610","931170240","814773960","465585120","174594420","42007680","6126120","480480","15015"],["208606320","1274816400","3432198000","5354228880","5354228880","3585005424","1629547920","498841200","99768240","12252240","816816","21840"],["882230895","5840976960","17210021400","29745716000","33463930500","25700298624","13742520792","5121436320","1309458150","221707200","23279256","1336608","30940"],["3712555350","26466926850","84694165920","160626866400","200783583000","174012438600","107084577600","47117214144","14724129420","3200897700","465585120","42325920","2116296","42840"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","5","72","693","5632","41805","293640","1988525","13125120","85015125","542842440","3427638885","21450332160","133262943645","822933425160"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["5"]],"recurrence":{"denominator":{"coeffs":[["-3"],["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["6","6"],["10","6"],["4","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-3","-3"],["0","-2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181808,"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-04639af180f64be3"]},{"id":181809,"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-04639af180f64be3"]},{"id":181810,"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-04639af180f64be3"]}],"region_exact":{},"incoming_edges":[{"id":"e-04639af180f64be3","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-ae7fa795b878b77c","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":"064674f9d6cccc3fbdb5a611e8312b209c9b462c4e9ea0d2200a66f9e2b0970d","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-a893b3f9716e8c07","role":"sequence","name":"shift_x_plus_one of Coefficient triangle of certain polynomials N(4; m,x).","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:       0;\n  1:       5;\n  2:      42,       30;\n  3:     252,      336,      105;\n  4:    1320,     2520,     1512,      280;\n  5:    6435,    15840,    13860,     5040,      630;\n  6:   30030,    90090,   102960,    55440,    13860,    1260;\n  7:  136136,   480480,   675675,   480480,   180180,   33264,    2310;\n  8:  604656,  2450448,  4084080,  3603600,  1801800,  504504,   72072,   3960;\n  9: 2645370, 12093120, 23279256, 24504480, 15315300, 5765760, 1261260, 144144, 6435;\n","recurrence_sentence":"The row polynomials satisfy (n^2+2*n-3)*P(n,x) = (4*n^2+10*n+6+(3*n^2+6*n+6)*x)*P(n-1,x) - ((2*n+3)*x+(2*n+3)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 5.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 5;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 10*n + 6 + (3*n^2 + 6*n + 6)*x)*P[n - 1, x] - ((2*n + 3)*x + (2*n + 3)*x^2)*D[P[n - 1, x], x])/(n^2 + 2*n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 5; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 + 10*n + 6 + (3*n^2 + 6*n + 6)*x)*v[i - 1] - ((2*n + 3)*x + (2*n + 3)*x^2)*deriv(v[i - 1], x))/(n^2 + 2*n - 3)); 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-9516b04042fe64c0","steps":["x \u21a6 x+1","D"],"edges":["e-06bdf5b2aa1ff3d5","e-04639af180f64be3"],"text":"A062264 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A062264 \u2192 x \u21a6 x+1 \u2192 D","chain":null}