{"id":"n-cf9aeea1b9c52e7c","revision":0,"first_index":0,"fingerprint":"f6-ddf2baab3dc16fba","name":"theta_plus_one_squared of basis_map_A013609 of Triangle of coefficients in expansion of (14 + x)^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":false,"rows":[["1"],["15","8"],["225","240","36"],["3375","5400","1620","128"],["50625","108000","48600","7680","400"],["759375","2025000","1215000","288000","30000","1152"],["11390625","36450000","27337500","8640000","1350000","103680","3136"],["170859375","637875000","574087500","226800000","47250000","5443200","329280","8192"],["2562890625","10935000000","11481750000","5443200000","1417500000","217728000","19756800","983040","20736"],["38443359375","184528125000","221433750000","122472000000","38272500000","7348320000","889056000","66355200","2799360","51200"],["576650390625","3075468750000","4151882812500","2624400000000","956812500000","220449600000","33339600000","3317760000","209952000","7680000","123904"],["8649755859375","50745234375000","76117851562500","54128250000000","22553437500000","6062364000000","1100206800000","136857600000","11547360000","633600000","20444160","294912"],["129746337890625","830376562500000","1370121328125000","1082565000000000","507452343750000","155889360000000","33006204000000","4926873600000","519631200000","38016000000","1839974400","53084160","692224"],["1946195068359375","13493619140625000","24288514453125000","21110017500000000","10994800781250000","3799803150000000","919458540000000","160123392000000","20265616800000","1853280000000","119598336000","5175705600","134983680","1605632"],["29192926025390625","217973847656250000","425049002929687500","403009425000000000","230890816406250000","88662073500000000","24135786675000000","4803701760000000","709296588000000","77837760000000","6278912640000","362299392000","14173286400","337182720","3686400"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","23","501","10523","215305","4318527","85274941","1662652547","32078829201","613456316135","11642539169029","219506159395947","4114643550816409","76734775832790287","1424511370026765645"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["225","-30","0"],["0","90","0"],["0","0","4"]]},"terms":[{"coeff":{"coeffs":[["3375","1350","180","8"],["0","1350","300","16"],["0","0","60","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189926,"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-753c2797fb3ee963"]},{"id":189927,"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-753c2797fb3ee963"]},{"id":189928,"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-753c2797fb3ee963"]},{"id":189929,"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-753c2797fb3ee963"]}],"region_exact":{},"incoming_edges":[{"id":"e-753c2797fb3ee963","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-cf9aeea1b9c52e7c","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"48f606a4b003e73aec52efbac400b148b45b45b2a75ef2cb0aa258703901302c","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-e91c3b4104e64a83","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients in expansion of (14 + x)^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":[],"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:          15,            8;\n  2:         225,          240,           36;\n  3:        3375,         5400,         1620,          128;\n  4:       50625,       108000,        48600,         7680,         400;\n  5:      759375,      2025000,      1215000,       288000,       30000,       1152;\n  6:    11390625,     36450000,     27337500,      8640000,     1350000,     103680,      3136;\n  7:   170859375,    637875000,    574087500,    226800000,    47250000,    5443200,    329280,     8192;\n  8:  2562890625,  10935000000,  11481750000,   5443200000,  1417500000,  217728000,  19756800,   983040,   20736;\n  9: 38443359375, 184528125000, 221433750000, 122472000000, 38272500000, 7348320000, 889056000, 66355200, 2799360, 51200;\n","recurrence_sentence":"The row polynomials satisfy (225+(90*n-30)*x+4*n^2*x^2)*P(n,x) = (3375+(1350*n+1350)*x+(60*n^2+300*n+180)*x^2+(8*n^2+16*n+8)*x^3)*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[((3375 + (1350*n + 1350)*x + (60*n^2 + 300*n + 180)*x^2 + (8*n^2 + 16*n + 8)*x^3)*P[n - 1, x])/(225 + (90*n - 30)*x + 4*n^2*x^2)]];\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] = ((3375 + (1350*n + 1350)*x + (60*n^2 + 300*n + 180)*x^2 + (8*n^2 + 16*n + 8)*x^3)*v[i - 1])/(225 + (90*n - 30)*x + 4*n^2*x^2)); 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-a679863398df5985","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-267357ec4c02d3cf","e-753c2797fb3ee963"],"text":"A147716 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A147716 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}