{"id":"n-af48e301204acbca","revision":0,"first_index":1,"fingerprint":"f6-3160782481402e19","name":"one_plus_derivative of basis_map_A013610 of T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","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":[["1"],["9","6"],["67","162","54"],["459","2016","2430","540"],["3021","19440","42390","34020","5670"],["19413","165150","527310","763560","459270","61236"],["122767","1300698","5492340","11634840","12604410","6062364","673596"],["767511","9740976","51254532","143564400","227338650","196690032","78810732","7505784"],["4757017","70376688","443836260","1549270800","3242803410","4098158064","2951024076","1013280840","84440070"],["29285793","494956710","3640465620","15233997240","39733744050","66349879596","69747497820","43008142320","12919330710","956987460"],["179320923","3408957090","28656368730","139886964360","437411724750","910027522404","1263598736400","1136901102480","612950468130","163644855660","10909657044"],["1093149027","23089393008","218404762290","1218801483900","4446173121390","11089125343296","19180325468304","22787785397520","17919111694770","8582263541280","2061925181316","124965162504"],["6639266661","154263522816","1621824486282","10186506029700","42486062919300","123624405672768","256313481632208","379263317477616","393702248575350","274923357508800","118446146526708","25867788638328","1437099368796"],["40197431277","1018979392158","11788168973490","82310025798000","386440149235140","1285671012922296","3109790467287504","5515219534659120","7129106314400070","6570286406463780","4125912287813316","1615299690526704","323347357979100","16581915793800"],["242720765223","6666247389690","84159155251080","646747630598160","3376042107738300","12644285240891304","34969430830534200","72348699846131280","112076897825422530","128598837016710060","106557085931239944","60789303300070800","21807982921479300","4029405537893400","191876454185400"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","15","283","5445","104541","1995939","37891015","715672617","13457947225","252114287319","4707575677971","87653063698605","1628033141625333","30172812508675755","558127662766300671"],"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":[["-75","-27","27"],["156","27","-27"],["-108","0","0"],["27","0","0"]]},"terms":[{"coeff":{"coeffs":[["-27","-45","81","162"],["108","207","-81","-162"],["-162","-324","0","0"],["81","162","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["28","90","-72","-324","-162"],["-72","-324","-324","0","0"],["36","162","162","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188928,"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-006a3124500472c5"]},{"id":188929,"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-006a3124500472c5"]},{"id":188930,"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-006a3124500472c5"]},{"id":188931,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-006a3124500472c5"]}],"region_exact":{},"incoming_edges":[{"id":"e-006a3124500472c5","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-af48e301204acbca","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":"0b2b8fadc4670085a1f0e525591bbb46725d8da6abeb6058d534f99e68755666","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-ecc015942a4103af","role":"sequence","name":"basis_map_A013610 of T(n,k) = binomial(n,k)*binomial(n+k,k), 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":[],"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:        9,         6;\n   3:       67,       162,         54;\n   4:      459,      2016,       2430,         540;\n   5:     3021,     19440,      42390,       34020,        5670;\n   6:    19413,    165150,     527310,      763560,      459270,       61236;\n   7:   122767,   1300698,    5492340,    11634840,    12604410,     6062364,      673596;\n   8:   767511,   9740976,   51254532,   143564400,   227338650,   196690032,    78810732,     7505784;\n   9:  4757017,  70376688,  443836260,  1549270800,  3242803410,  4098158064,  2951024076,  1013280840,    84440070;\n  10: 29285793, 494956710, 3640465620, 15233997240, 39733744050, 66349879596, 69747497820, 43008142320, 12919330710, 956987460;\n","recurrence_sentence":"The row polynomials satisfy (27*n^3-108*n^2+156*n-75+(27*n-27)*x-(27*n-27)*x^2)*P(n,x) = (81*n^3-162*n^2+108*n-27+(162*n^3-324*n^2+207*n-45)*x-(81*n-81)*x^2-(162*n-162)*x^3)*P(n-1,x) + (36*n^2-72*n+28+(162*n^2-324*n+90)*x+(162*n^2-324*n-72)*x^2-324*x^3-162*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[((81*n^3 - 162*n^2 + 108*n - 27 + (162*n^3 - 324*n^2 + 207*n - 45)*x - (81*n - 81)*x^2 - (162*n - 162)*x^3)*P[n - 1, x] + (36*n^2 - 72*n + 28 + (162*n^2 - 324*n + 90)*x + (162*n^2 - 324*n - 72)*x^2 - 324*x^3 - 162*x^4)*D[P[n - 1, x], x])/(27*n^3 - 108*n^2 + 156*n - 75 + (27*n - 27)*x - (27*n - 27)*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] = ((81*n^3 - 162*n^2 + 108*n - 27 + (162*n^3 - 324*n^2 + 207*n - 45)*x - (81*n - 81)*x^2 - (162*n - 162)*x^3)*v[i - 1] + (36*n^2 - 72*n + 28 + (162*n^2 - 324*n + 90)*x + (162*n^2 - 324*n - 72)*x^2 - 324*x^3 - 162*x^4)*deriv(v[i - 1], x))/(27*n^3 - 108*n^2 + 156*n - 75 + (27*n - 27)*x - (27*n - 27)*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-4689fb1386d440ad","steps":["basis A013610","D+1"],"edges":["e-50c3f3c57fe6951a","e-006a3124500472c5"],"text":"A063007 \u2192 basis A013610 \u2192 D+1"},"headline":"A063007 \u2192 basis A013610 \u2192 D+1","chain":null}