{"id":"n-f44c764f0d48352a","revision":0,"first_index":0,"fingerprint":"f6-bd3dad0f6f053f13","name":"one_plus_derivative of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","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"],["11","3"],["112","66","9"],["1088","1008","297","27"],["10240","13056","6048","1188","81"],["94208","153600","97920","30240","4455","243"],["851968","1695744","1382400","587520","136080","16038","729"],["7602176","17891328","17805312","9676800","3084480","571536","56133","2187"],["67108864","182452224","214695936","142442496","58060800","14805504","2286144","192456","6561"],["587202560","1811939328","2463105024","1932263424","961486848","313528320","66624768","8817984","649539","19683"],["5100273664","17616076800","27179089920","24631050240","14491975680","5768921088","1567641600","285534720","33067440","2165130","59049"],["44023414784","168309030912","290665267200","298969989120","203206164480","95647039488","31729065984","7390310400","1177830720","121247280","7144929","177147"],["377957122048","1584842932224","3029562556416","3487983206400","2690729902080","1463084384256","573882236928","163178053632","33256396800","4711322880","436490208","23383404","531441"],["3229815406592","14740327759872","30904437178368","39384313233408","34007836262400","20987693236224","9510048497664","3197343891456","795493011456","144111052800","18374159232","1547556192","75996063","1594323"],["27487790694400","135652247076864","309546882957312","432662120497152","413535288950784","285665824604160","146913852653568","57060290985984","16786055430144","3712300720128","605266421760","70155880704","5416446672","245525742","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","14","187","2420","30613","380666","4670479","56689952","682050985","8145637478","96675855331","1141246682444","13409648518717","156921418836050","1829703743628343"],"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":[["5","3"],["3","0"]]},"terms":[{"coeff":{"coeffs":[["64","48","9"],["24","9","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":186019,"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-5266cf205eb085fc"]},{"id":186020,"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-5266cf205eb085fc"]},{"id":186021,"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-5266cf205eb085fc"]},{"id":186022,"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-5266cf205eb085fc"]}],"region_exact":{},"incoming_edges":[{"id":"e-5266cf205eb085fc","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-f44c764f0d48352a","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":"0ae34299358d2fc98d93ac59ab8d372a3f310d2573b081b7652bbf1b5c528c1b","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-a227586ec36efe06","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","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-dde8354176f54804","kind":"fingerprint_related","node_a_id":"n-f44c764f0d48352a","node_b_id":"n-9f5c546bbf0f2937","fingerprint":"f6-bd3dad0f6f053f13","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-bd3dad0f6f053f13"},"other_id":"n-9f5c546bbf0f2937","other":{"id":"n-9f5c546bbf0f2937","name":"one_plus_derivative of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.","first_index":0,"fingerprint":"f6-bd3dad0f6f053f13","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["112","66","9"],["1088","1008","297","27"],["10240","13056","6048","1188","81"]],"card_rows":[["112","66","9"],["1088","1008","297","27"],["10240","13056","6048","1188","81"],["94208","153600","97920","30240","4455","243"]],"row_sums_preview":["187","2420","30613","380666","4670479","56689952","682050985","8145637478"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-108b34f10b7468f4","steps":["D+1"],"edges":["e-4841f66f25363e11"],"text":"A038281 \u2192 D+1"},"headline":"A038281 \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:        11,          3;\n  2:       112,         66,          9;\n  3:      1088,       1008,        297,         27;\n  4:     10240,      13056,       6048,       1188,        81;\n  5:     94208,     153600,      97920,      30240,      4455,       243;\n  6:    851968,    1695744,    1382400,     587520,    136080,     16038,      729;\n  7:   7602176,   17891328,   17805312,    9676800,   3084480,    571536,    56133,    2187;\n  8:  67108864,  182452224,  214695936,  142442496,  58060800,  14805504,  2286144,  192456,   6561;\n  9: 587202560, 1811939328, 2463105024, 1932263424, 961486848, 313528320, 66624768, 8817984, 649539, 19683;\n","recurrence_sentence":"The row polynomials satisfy (3*n+5+3*x)*P(n,x) = (24*n+64+(9*n+48)*x+9*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[((24*n + 64 + (9*n + 48)*x + 9*x^2)*P[n - 1, x])/(3*n + 5 + 3*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] = ((24*n + 64 + (9*n + 48)*x + 9*x^2)*v[i - 1])/(3*n + 5 + 3*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-f211cb6153c37c6d","steps":["basis A013610","D+1"],"edges":["e-892638ce5a781886","e-5266cf205eb085fc"],"text":"A027466 \u2192 basis A013610 \u2192 D+1"},"headline":"A027466 \u2192 basis A013610 \u2192 D+1","chain":null}