{"id":"n-7a1b67fb7e683084","revision":0,"first_index":0,"fingerprint":"f6-2b0c83ee9d9e3a42","name":"theta_plus_one_squared of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(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":false,"rows":[["1"],["7","8"],["49","112","36"],["343","1176","756","128"],["2401","10976","10584","3584","400"],["16807","96040","123480","62720","14000","1152"],["117649","806736","1296540","878080","294000","48384","3136"],["823543","6588344","12706092","10756480","4802000","1185408","153664","8192"],["5764801","52706752","118590192","120472576","67228000","22127616","4302592","458752","20736"],["40353607","415065672","1067311728","1264962048","847072800","348509952","90354432","14450688","1306368","51200"],["282475249","3228288560","9338977620","12649620480","9882516000","4879139328","1581202560","337182720","45722880","3584000","123904"],["1977326743","24857821912","79900141860","121752597120","108707676000","62615621376","24350519424","6490767360","1173553920","137984000","9540608","294912"],["13841287201","189823367328","671161191624","1136357573120","1141430598000","751387456512","340907271936","109044891648","24644632320","3863552000","400705536","24772608","692224"],["96889010407","1439493868904","5552333494344","10340853915392","11541131602000","8547032317824","4431794535168","1653847523328","448532308224","87895808000","12154734592","1127153664","62992384","1605632"],["678223072849","10851569165584","45344056870476","92127607609856","113103089699600","93067685238528","54289483055808","23153865326592","7326027700992","1722757836800","297790997504","36820353024","3086626816","157351936","3686400"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","15","197","2403","27945","314199","3444525","37023723","391672017","4089438495","42228833301","431973845235","4382887992057","44153150869863","442002224592765"],"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":[["49","-14","0"],["0","42","0"],["0","0","4"]]},"terms":[{"coeff":{"coeffs":[["343","294","84","8"],["0","294","140","16"],["0","0","28","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184312,"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-3df8e607522c4e5a"]},{"id":184313,"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-3df8e607522c4e5a"]},{"id":184314,"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-3df8e607522c4e5a"]},{"id":184315,"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-3df8e607522c4e5a"]}],"region_exact":{},"incoming_edges":[{"id":"e-3df8e607522c4e5a","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-7a1b67fb7e683084","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":"fd7239b7b586d3ca3171e38b2b58269ec950bce808fd539a21d974db31aec23e","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-6a4f4798749d2440","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(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-b40e4f80eba8423d","kind":"fingerprint_related","node_a_id":"n-7a1b67fb7e683084","node_b_id":"n-906b14404b065035","fingerprint":"f6-2b0c83ee9d9e3a42","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2b0c83ee9d9e3a42"},"other_id":"n-906b14404b065035","other":{"id":"n-906b14404b065035","name":"theta_plus_one_squared of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-2b0c83ee9d9e3a42","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["196","448","144"],["2744","9408","6048","1024"],["38416","175616","169344","57344","6400"]],"card_rows":[["196","448","144"],["2744","9408","6048","1024"],["38416","175616","169344","57344","6400"],["537824","3073280","3951360","2007040","448000","36864"]],"row_sums_preview":["788","19224","447120","10054368","220449600","4739036544","100268036352","2093792509440"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-ccf7b0982517b62a","e-aad5905b00ba5949"],"text":"A038328 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A038328 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","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:        7,         8;\n  2:       49,       112,         36;\n  3:      343,      1176,        756,        128;\n  4:     2401,     10976,      10584,       3584,       400;\n  5:    16807,     96040,     123480,      62720,     14000,      1152;\n  6:   117649,    806736,    1296540,     878080,    294000,     48384,     3136;\n  7:   823543,   6588344,   12706092,   10756480,   4802000,   1185408,   153664,     8192;\n  8:  5764801,  52706752,  118590192,  120472576,  67228000,  22127616,  4302592,   458752,   20736;\n  9: 40353607, 415065672, 1067311728, 1264962048, 847072800, 348509952, 90354432, 14450688, 1306368, 51200;\n","recurrence_sentence":"The row polynomials satisfy (49+(42*n-14)*x+4*n^2*x^2)*P(n,x) = (343+(294*n+294)*x+(28*n^2+140*n+84)*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[((343 + (294*n + 294)*x + (28*n^2 + 140*n + 84)*x^2 + (8*n^2 + 16*n + 8)*x^3)*P[n - 1, x])/(49 + (42*n - 14)*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] = ((343 + (294*n + 294)*x + (28*n^2 + 140*n + 84)*x^2 + (8*n^2 + 16*n + 8)*x^3)*v[i - 1])/(49 + (42*n - 14)*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-22693abd431a71f2","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-4c464713c83c5f6b","e-3df8e607522c4e5a"],"text":"A038255 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A038255 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}
