{"id":"n-0848a50e390b197e","revision":0,"first_index":0,"fingerprint":"f6-1d70f6b910bc9524","name":"derivative of Triangle of unsigned 2-Lah numbers.","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":[["10","2"],["90","36","3"],["840","504","84","4"],["8400","6720","1680","160","5"],["90720","90720","30240","4320","270","6"],["1058400","1270080","529200","100800","9450","420","7"],["13305600","18627840","9313920","2217600","277200","18480","616","8"],["179625600","287400960","167650560","47900160","7484400","665280","33264","864","9"],["2594592000","4670265600","3113510400","1037836800","194594400","21621600","1441440","56160","1170","10"],["39956716800","79913433600","59935075200","22832409600","4994589600","665945280","55495440","2882880","90090","1540","11"],["653837184000","1438441804800","1198701504000","513729216000","128432304000","19978358400","1997835840","129729600","5405400","138600","1980","12"],["11333177856000","27199626854400","24932991283200","11872852992000","3339239904000","593642649600","69258309120","5396751360","281080800","9609600","205920","2496","13"],["207484333056000","539459265945600","539459265945600","282573901209600","88304344128000","17660868825600","2354782510080","214071137280","13379446080","571771200","16336320","297024","3094","14"],["4001483566080000","11204153985024000","12137833483776000","6935904847872000","2384217291456000","529826064768000","79473909715200","8257029580800","602075073600","30875644800","1102701600","26732160","417690","3780","15"],["81096733605888000","243290200817664000","283838567620608000","175709589479424000","65891096054784000","16106712368947200","2684452061491200","313767124070400","26147260339200","1564366003200","67044257280","2031644160","42325920","574560","4560","16"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["12","129","1432","16965","216276","2968357","43761264","690761097","11633919580","208356640041","3955253482632","79346477498509","1677524800611492","37281784258845645","868958899878025696"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["10","2"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["10","2"],["7","1"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2"],["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189964,"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-98550cc090b99e33"]},{"id":189965,"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-98550cc090b99e33"]},{"id":189966,"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-98550cc090b99e33"]}],"region_exact":{},"incoming_edges":[{"id":"e-98550cc090b99e33","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-0848a50e390b197e","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":"ad41cbd2070bba4654fec17442065217d404452510c4e86ce8a0559969e6a3de","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-f3352d3240f9f08b","role":"sequence","name":"Triangle of unsigned 2-Lah numbers.","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-70b247f70a15d659","kind":"fingerprint_related","node_a_id":"n-0848a50e390b197e","node_b_id":"n-1d88497fc221d044","fingerprint":"f6-1d70f6b910bc9524","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1d70f6b910bc9524"},"other_id":"n-1d88497fc221d044","other":{"id":"n-1d88497fc221d044","name":"one_plus_derivative of Triangle of unsigned 2-Lah numbers.","first_index":0,"fingerprint":"f6-1d70f6b910bc9524","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["30","12","1"],["210","126","21","1"],["1680","1344","336","32","1"]],"card_rows":[["30","12","1"],["210","126","21","1"],["1680","1344","336","32","1"],["15120","15120","5040","720","45","1"]],"row_sums_preview":["43","358","3393","36046","424051","5470158","76751233","1163391958"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f3352d3240f9f08b","steps":["D+1"],"edges":["e-138ea761723e80e6"],"text":"A143497 \u2192 D+1"},"headline":"A143497 \u2192 D+1","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:          10,           2;\n  1:          90,          36,           3;\n  2:         840,         504,          84,           4;\n  3:        8400,        6720,        1680,         160,          5;\n  4:       90720,       90720,       30240,        4320,        270,         6;\n  5:     1058400,     1270080,      529200,      100800,       9450,       420,        7;\n  6:    13305600,    18627840,     9313920,     2217600,     277200,     18480,      616,       8;\n  7:   179625600,   287400960,   167650560,    47900160,    7484400,    665280,    33264,     864,     9;\n  8:  2594592000,  4670265600,  3113510400,  1037836800,  194594400,  21621600,  1441440,   56160,  1170,   10;\n  9: 39956716800, 79913433600, 59935075200, 22832409600, 4994589600, 665945280, 55495440, 2882880, 90090, 1540, 11;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = (n^2+7*n+10+(n+2)*x)*P(n-1,x) + (n+2)*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 10 + 2*x.","mathematica":"P[0, x_] := 10 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + 7*n + 10 + (n + 2)*x)*P[n - 1, x] + (n + 2)*x*D[P[n - 1, x], x])/(n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 10 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 7*n + 10 + (n + 2)*x)*v[i - 1] + (n + 2)*x*deriv(v[i - 1], x))/(n + 1)); 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-f3352d3240f9f08b","steps":["D"],"edges":["e-98550cc090b99e33"],"text":"A143497 \u2192 D"},"headline":"A143497 \u2192 D","chain":null}