{"id":"n-e6ca5cf785756a3f","revision":0,"first_index":1,"fingerprint":"f6-8449ee3a266702aa","name":"one_plus_derivative of Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A286718 (|S1hat[3,1]| generalized Stirling 1), for n >= 0.","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"],["3","2"],["23","27","4"],["250","516","171","8"],["3474","11450","6789","951","16"],["58664","282156","271180","71878","4995","32"],["1165576","7644912","10956816","4785412","681519","25551","64"],["26632752","226615016","453098292","299180520","71921390","6070368","129099","128"],["687950000","7315140432","19382702668","18109844564","6667956150","980938510","52062953","648567","256"],["19822519360","255937349952","863843561784","1084479861048","571396220388","130661739900","12567234288","435999546","3249747","512"],["630321908160","9661001187008","40273646954904","65235547942920","46677627654420","15434255988396","2348089186212","154350938616","3595339395","16264095","1024"],["21925134573440","391732805316096","1968020535193600","3983325933985824","3714237165217680","1683161518606008","374257531732056","39714689717640","1840857970386","29348478580","81354267","2048"],["828134675025280","16993010356279680","100863692208306432","248625567115730240","292342178871302544","174098644459005888","53507550502141512","8395406467465752","642953327925690","21499966757186","237961275837","406845063","4096"],["33754030376371200","785663592607136000","5420587828638214272","15936000490575431232","23010613806296472112","17414238905803895904","7086620871268332336","1549849134469110384","177754846552364016","10077707434259628","247319760617236","1920779223054","2034385059","8192"],["1476676808634137600","38583303370076889600","305247483787444985728","1052021271276232296064","1825426759203038609712","1708448812248347284016","889562366194824046704","259421207432056941456","41832188869409760912","3602262147069641464","154160132898145944","2813398287093356","15457784732951","10172269359","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","54","945","22680","688905","25259850","1083647565","53197244100","2939147736525","180418453365150","12178245602147625","896318876318065200","71425410456595820625","6125779320336276851250"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["3","2"]],"recurrence":{"denominator":{"coeffs":[["-10","6"],["3","0"]]},"terms":[{"coeff":{"coeffs":[["280","1048","-635","-549","126"],["-393","-1137","144","432","-54"],["180","423","108","-81","0"],["-27","-54","-27","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-168","102","309","-297","54"],["0","135","-36","-180","81","0"],["0","-27","0","27","0","0"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["76","110","-141","18"],["-81","-75","45","0"],["18","18","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-30","48","-18"],["0","9","-9","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190731,"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-df865e6b724301e1"]},{"id":190732,"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-df865e6b724301e1"]},{"id":190733,"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-df865e6b724301e1"]},{"id":190734,"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-df865e6b724301e1"]}],"region_exact":{},"incoming_edges":[{"id":"e-df865e6b724301e1","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-e6ca5cf785756a3f","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":"bb8536a39ba63daa05870a93c44ce4e07bbc5a84f2c40e0c32364eb5800413d1","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-d0e888e78d074358","role":"sequence","name":"Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A286718 (|S1hat[3,1]| generalized Stirling 1), for n >= 0.","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:           3,            2;\n   3:          23,           27,            4;\n   4:         250,          516,          171,             8;\n   5:        3474,        11450,         6789,           951,           16;\n   6:       58664,       282156,       271180,         71878,         4995,           32;\n   7:     1165576,      7644912,     10956816,       4785412,       681519,        25551,          64;\n   8:    26632752,    226615016,    453098292,     299180520,     71921390,      6070368,      129099,       128;\n   9:   687950000,   7315140432,  19382702668,   18109844564,   6667956150,    980938510,    52062953,    648567,     256;\n  10: 19822519360, 255937349952, 863843561784, 1084479861048, 571396220388, 130661739900, 12567234288, 435999546, 3249747, 512;\n","recurrence_sentence":"The row polynomials satisfy (3*n-10+6*x)*P(n,x) = -(27*n^3-180*n^2+393*n-280+(54*n^3-423*n^2+1137*n-1048)*x+(27*n^3-108*n^2-144*n+635)*x^2+(81*n^2-432*n+549)*x^3+(54*n-126)*x^4)*P(n-2,x) - ((27*n^2-135*n+168)*x+(36*n-102)*x^2-(27*n^2-180*n+309)*x^3-(81*n-297)*x^4-54*x^5)*d/dx P(n-2,x) + (18*n^2-81*n+76+(18*n^2-75*n+110)*x+(45*n-141)*x^2+18*x^3)*P(n-1,x) + ((9*n-30)*x-(9*n-48)*x^2-18*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(27*n^3 - 180*n^2 + 393*n - 280 + (54*n^3 - 423*n^2 + 1137*n - 1048)*x + (27*n^3 - 108*n^2 - 144*n + 635)*x^2 + (81*n^2 - 432*n + 549)*x^3 + (54*n - 126)*x^4)*P[n - 2, x] - ((27*n^2 - 135*n + 168)*x + (36*n - 102)*x^2 - (27*n^2 - 180*n + 309)*x^3 - (81*n - 297)*x^4 - 54*x^5)*D[P[n - 2, x], x] + (18*n^2 - 81*n + 76 + (18*n^2 - 75*n + 110)*x + (45*n - 141)*x^2 + 18*x^3)*P[n - 1, x] + ((9*n - 30)*x - (9*n - 48)*x^2 - 18*x^3)*D[P[n - 1, x], x])/(3*n - 10 + 6*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 2*x; for(n = 3, N, my(i = n); v[i] = (-(27*n^3 - 180*n^2 + 393*n - 280 + (54*n^3 - 423*n^2 + 1137*n - 1048)*x + (27*n^3 - 108*n^2 - 144*n + 635)*x^2 + (81*n^2 - 432*n + 549)*x^3 + (54*n - 126)*x^4)*v[i - 2] - ((27*n^2 - 135*n + 168)*x + (36*n - 102)*x^2 - (27*n^2 - 180*n + 309)*x^3 - (81*n - 297)*x^4 - 54*x^5)*deriv(v[i - 2], x) + (18*n^2 - 81*n + 76 + (18*n^2 - 75*n + 110)*x + (45*n - 141)*x^2 + 18*x^3)*v[i - 1] + ((9*n - 30)*x - (9*n - 48)*x^2 - 18*x^3)*deriv(v[i - 1], x))/(3*n - 10 + 6*x)); 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-d0e888e78d074358","steps":["D+1"],"edges":["e-df865e6b724301e1"],"text":"A290595 \u2192 D+1"},"headline":"A290595 \u2192 D+1","chain":null}