{"id":"n-cec7560ed0614a99","revision":0,"first_index":0,"fingerprint":"f6-7cc26d5446b1ed33","name":"derivative of derivative of Triangle of Dyck paths counted by number of long interior inclines.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],[],["4"],["30","12"],["120","144","24"],["350","840","420","40"],["840","3360","3360","960","60"],["1764","10584","17640","10080","1890","84"],["3360","28224","70560","67200","25200","3360","112"],["5940","66528","232848","332640","207900","55440","5544","144"],["9900","142560","665280","1330560","1247400","554400","110880","8640","180"],["15730","283140","1698840","4530240","5945940","3963960","1321320","205920","12870","220"],["24024","528528","3963960","13590720","23783760","22198176","11099088","2882880","360360","18480","264"],["35490","936936","8588580","36808200","82818450","103062960","72144072","28108080","5855850","600600","25740","312"],["50960","1589952","17489472","91611520","257657400","412251840","384768384","209873664","65585520","11211200","960960","34944","364"]],"degrees":[-1,-1,0,1,2,3,4,5,6,7,8,9,10,11,12],"row_sums":["0","0","4","42","288","1650","8580","42042","198016","906984","4069800","17978180","78450240","338985270","1453086180"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["4"]],"recurrence":{"denominator":{"coeffs":[["6"],["-7"],["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["6","-6"],["5","-3"],["1","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["0","1","-1"],["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175998,"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-feae821ba134d505"]},{"id":175999,"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-feae821ba134d505"]},{"id":176000,"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-feae821ba134d505"]}],"region_exact":{},"incoming_edges":[{"id":"e-feae821ba134d505","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-cec7560ed0614a99","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":"90d7bb00d07e7ff1328ae121f3fe612dadbd26bf40214f84fe65a7b14ab5ab2b","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-aa9a15392d2524ca","role":"sequence","name":"derivative of Triangle of Dyck paths counted by number of long interior inclines.","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":[{"id":"l-66516f80f02d1963","kind":"fingerprint_related","node_a_id":"n-c5aca7d2e34bc886","node_b_id":"n-cec7560ed0614a99","fingerprint":"f6-7cc26d5446b1ed33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7cc26d5446b1ed33"},"other_id":"n-c5aca7d2e34bc886","other":{"id":"n-c5aca7d2e34bc886","name":"derivative of Coefficient triangle of certain polynomials N(4; m,x).","first_index":0,"fingerprint":"f6-7cc26d5446b1ed33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["5"],["12","30"]],"card_rows":[["5"],["12","30"],["21","126","105"],["32","336","672","280"]],"row_sums_preview":["0","5","42","252","1320","6435","30030","136136"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9516b04042fe64c0","steps":["D"],"edges":["e-3c18fb3613f81561"],"text":"A062264 \u2192 D"},"headline":"A062264 \u2192 D","chain":null}}],"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:    0;\n  2:    4;\n  3:   30,    12;\n  4:  120,   144,     24;\n  5:  350,   840,    420,     40;\n  6:  840,  3360,   3360,    960,     60;\n  7: 1764, 10584,  17640,  10080,   1890,    84;\n  8: 3360, 28224,  70560,  67200,  25200,  3360,  112;\n  9: 5940, 66528, 232848, 332640, 207900, 55440, 5544, 144;\n","recurrence_sentence":"The row polynomials satisfy (n^3-7*n+6)*P(n,x) = (n^3+5*n^2+6*n+(3*n^3-3*n^2-6*n)*x)*P(n-1,x) + ((2*n^2+n)*x-(2*n^2+n)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 + 5*n^2 + 6*n + (3*n^3 - 3*n^2 - 6*n)*x)*P[n - 1, x] + ((2*n^2 + n)*x - (2*n^2 + n)*x^2)*D[P[n - 1, x], x])/(n^3 - 7*n + 6)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 4; for(n = 3, N, my(i = n + 1); v[i] = ((n^3 + 5*n^2 + 6*n + (3*n^3 - 3*n^2 - 6*n)*x)*v[i - 1] + ((2*n^2 + n)*x - (2*n^2 + n)*x^2)*deriv(v[i - 1], x))/(n^3 - 7*n + 6)); 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-c963abfc0266d66c","steps":["D","D"],"edges":["e-0041de9aacba5ab3","e-feae821ba134d505"],"text":"A108838 \u2192 D \u2192 D"},"headline":"A108838 \u2192 D \u2192 D","chain":null}