{"id":"n-3470a64b61130b24","revision":0,"first_index":0,"fingerprint":"f6-354a4e768d29be07","name":"one_plus_derivative of basis_map_A007318 of Triangle of coefficients in expansion of (3+5x)^n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["13","5"],["144","130","25"],["1472","2160","975","125"],["14336","29440","21600","6500","625"],["135168","358400","368000","180000","40625","3125"],["1245184","4055040","5376000","3680000","1350000","243750","15625"],["11272192","43581440","70963200","62720000","32200000","9450000","1421875","78125"],["100663296","450887680","871628800","946176000","627200000","257600000","63000000","8125000","390625"],["889192448","4529848320","10144972800","13074432000","10644480000","5644800000","1932000000","405000000","45703125","1953125"],["7784628224","44459622400","113246208000","169082880000","163430400000","106444800000","47040000000","13800000000","2531250000","253906250","9765625"],["67645734912","428154552320","1222639616000","2076180480000","2324889600000","1797734400000","975744000000","369600000000","94875000000","15468750000","1396484375","48828125"],["584115552256","4058744094720","12844636569600","24452792320000","31142707200000","27898675200000","17977344000000","8363520000000","2772000000000","632500000000","92812500000","7617187500","244140625"],["5016521801728","37967510896640","131909183078400","278300459008000","397357875200000","404855193600000","302235648000000","166932480000000","67953600000000","20020000000000","4111250000000","548437500000","41259765625","1220703125"],["42880953483264","351156526120960","1328862881382400","3077880938496000","4870258032640000","5563010252800000","4723310592000000","3022356480000000","1460659200000000","528528000000000","140140000000000","26162500000000","3199218750000","222167968750","6103515625"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","18","299","4732","72501","1085318","15965599","231686832","3325671401","47312381818","668083460499","9374377445732","130827708764701","1817250639553518","25138633847156999"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["3","5"],["5","0"]]},"terms":[{"coeff":{"coeffs":[["64","80","25"],["40","25","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187292,"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-5712cc1fe665ed0e"]},{"id":187293,"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-5712cc1fe665ed0e"]},{"id":187294,"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-5712cc1fe665ed0e"]},{"id":187295,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-5712cc1fe665ed0e"]}],"region_exact":{},"incoming_edges":[{"id":"e-5712cc1fe665ed0e","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-3470a64b61130b24","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":"5fce068e7fad1ccb6abe9eba8f0a1183d7676359c004f325909b90a259b85a06","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-93086473ff31e9a0","role":"sequence","name":"basis_map_A007318 of Triangle of coefficients in expansion of (3+5x)^n.","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-94196de85166780a","kind":"fingerprint_related","node_a_id":"n-3470a64b61130b24","node_b_id":"n-dd032736ed5eddee","fingerprint":"f6-354a4e768d29be07","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-354a4e768d29be07"},"other_id":"n-dd032736ed5eddee","other":{"id":"n-dd032736ed5eddee","name":"one_plus_derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","first_index":0,"fingerprint":"f6-354a4e768d29be07","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["576","520","100"],["11776","17280","7800","1000"],["229376","471040","345600","104000","10000"]],"card_rows":[["576","520","100"],["11776","17280","7800","1000"],["229376","471040","345600","104000","10000"],["4325376","11468800","11776000","5760000","1300000","100000"]],"row_sums_preview":["1196","37856","1160016","34730176","1021798336","29655914496","851371878656","24223939490816"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2bc1923905556bf6","steps":["basis A013609","D+1"],"edges":["e-772e88d8a50e6734","e-87fdf20e709b3a09"],"text":"A038319 \u2192 basis A013609 \u2192 D+1"},"headline":"A038319 \u2192 basis A013609 \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:         1;\n  1:        13,          5;\n  2:       144,        130,          25;\n  3:      1472,       2160,         975,         125;\n  4:     14336,      29440,       21600,        6500,         625;\n  5:    135168,     358400,      368000,      180000,       40625,       3125;\n  6:   1245184,    4055040,     5376000,     3680000,     1350000,     243750,      15625;\n  7:  11272192,   43581440,    70963200,    62720000,    32200000,    9450000,    1421875,     78125;\n  8: 100663296,  450887680,   871628800,   946176000,   627200000,  257600000,   63000000,   8125000,   390625;\n  9: 889192448, 4529848320, 10144972800, 13074432000, 10644480000, 5644800000, 1932000000, 405000000, 45703125, 1953125;\n","recurrence_sentence":"The row polynomials satisfy (5*n+3+5*x)*P(n,x) = (40*n+64+(25*n+80)*x+25*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((40*n + 64 + (25*n + 80)*x + 25*x^2)*P[n - 1, x])/(5*n + 3 + 5*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((40*n + 64 + (25*n + 80)*x + 25*x^2)*v[i - 1])/(5*n + 3 + 5*x)); 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-c44f7114310405ad","steps":["basis A007318","D+1"],"edges":["e-31008ec5d02c6da5","e-5712cc1fe665ed0e"],"text":"A013622 \u2192 basis A007318 \u2192 D+1"},"headline":"A013622 \u2192 basis A007318 \u2192 D+1","chain":null}
