{"id":"n-1fffa59090d6ab2a","revision":0,"first_index":0,"fingerprint":"f6-dc876aec23f20348","name":"theta_plus_one_squared of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["400","3840","5184"],["8000","115200","311040","221184"],["160000","3072000","12441600","17694720","8294400"],["3200000","76800000","414720000","884736000","829440000","286654464"],["64000000","1843200000","12441600000","35389440000","49766400000","34398535680","9364045824"],["1280000000","43008000000","348364800000","1238630400000","2322432000000","2407897497600","1310966415360","293534171136"],["25600000000","983040000000","9289728000000","39636172800000","92897280000000","128421199872000","104877313228800","46965467381760","8916100448256"],["512000000000","22118400000000","238878720000000","1189085184000000","3344302080000000","5778953994240000","6292638793728000","4226892064358400","1604898080686080","264180754022400"],["10240000000000","491520000000000","5971968000000000","33973862400000000","111476736000000000","231158159769600000","314631939686400000","281792804290560000","160489808068608000","52836150804480000","7671809096810496"],["204800000000000","10813440000000000","145981440000000000","934281216000000000","3503554560000000000","8475799191552000000","13843805346201600000","15498604235980800000","11769252591697920000","5811976588492800000","1687798001298309120","219122084616339456"],["4096000000000000","235929600000000000","3503554560000000000","24914165760000000000","105106636800000000000","290598829424640000000","553752213848064000000","743933003327078400000","706155155501875200000","464958127079424000000","202535760155797094400","52589300307921469440","6171938716693561344"],["81920000000000000","5111808000000000000","82811289600000000000","647768309760000000000","3036413952000000000000","9444461956300800000000","20567939371499520000000","32237096810840064000000","36720068086097510400000","30222278260162560000000","17553099213502414848000","6836609040029791027200","1604704066340325949440","171791714339210133504"],["1638400000000000000","110100480000000000000","1932263424000000000000","16488647884800000000000","85019590656000000000000","293827705307136000000000","719877878002483200000000","1289483872433602560000000","1713603177351217152000000","1692447582569103360000000","1228716944945169039360000","638083510402780495872000","224658569287645632921600","48101680014978837381120","4733037027712932249600"],["32768000000000000000","2359296000000000000000","44590694400000000000000","412216197120000000000000","2318716108800000000000000","8814831159214080000000000","23995929266749440000000000","48355645216260096000000000","73440136172195020800000000","84622379128455168000000000","73723016696710142361600000","47856263280208537190400000","22465856928764563292160000","7215252002246825607168000","1419911108313879674880000","129243464436747803295744"],["655360000000000000000","50331648000000000000000","1019215872000000000000000","10146860236800000000000000","61832429568000000000000000","256431451904409600000000000","767869736535982080000000000","1719311829911470080000000000","2937605446887800832000000000","3868451617300807680000000000","3931894223824540925952000000","3062800849933346380185600000","1797268554301165063372800000","769626880239661398097920000","227185777330220747980800000","41357908619759297054638080","3501690114583135795544064"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["9424","655424","41662720","2495550464","143267221504","7966113284096","432011901730816","22962460071034880","1200504998116458496","61901193495839768576","3154458711081493725184","159130235798472196358144","7957086198186629209784320","394816379487874500729503744","19456355459588547220439302144"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["400","3840","5184"]],"recurrence":{"denominator":{"coeffs":[["25","150","144"],["0","90","144"],["0","0","36"]]},"terms":[{"coeff":{"coeffs":[["500","5400","12240","7776"],["0","1800","6480","5184"],["0","0","720","864"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194950,"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-8784f281433f7aa1"]},{"id":194951,"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-8784f281433f7aa1"]},{"id":194952,"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-8784f281433f7aa1"]},{"id":194953,"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-8784f281433f7aa1"]}],"region_exact":{},"incoming_edges":[{"id":"e-8784f281433f7aa1","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-1fffa59090d6ab2a","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":"1beb211c5c3399d847e6e2acfd32f62bce763d444eb928325d3eaec61111e195","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-6081e8199e2268e3","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^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-ec8fae2106c84e51","kind":"fingerprint_related","node_a_id":"n-272e88c4cfcd9298","node_b_id":"n-1fffa59090d6ab2a","fingerprint":"f6-dc876aec23f20348","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dc876aec23f20348"},"other_id":"n-272e88c4cfcd9298","other":{"id":"n-272e88c4cfcd9298","name":"theta_plus_one_squared of basis_map_A013609 of Triangle of coefficients in expansion of (2+3x)^n.","first_index":0,"fingerprint":"f6-dc876aec23f20348","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5","24"],["25","240","324"]],"card_rows":[["1"],["5","24"],["25","240","324"],["125","1800","4860","3456"]],"row_sums_preview":["1","29","589","10241","162745","2437061","34977349","486212969"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-01cea33b6485d9cc","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-32b2129ae50cd22c","e-543e3e49cb8c0843"],"text":"A038220 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A038220 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-f87ff858ccc9fdc7","kind":"fingerprint_related","node_a_id":"n-d6ec43e2b2ecc097","node_b_id":"n-1fffa59090d6ab2a","fingerprint":"f6-dc876aec23f20348","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dc876aec23f20348"},"other_id":"n-d6ec43e2b2ecc097","other":{"id":"n-d6ec43e2b2ecc097","name":"theta_plus_one_squared of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.","first_index":0,"fingerprint":"f6-dc876aec23f20348","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["100","960","1296"],["1000","14400","38880","27648"],["10000","192000","777600","1105920","518400"]],"card_rows":[["100","960","1296"],["1000","14400","38880","27648"],["10000","192000","777600","1105920","518400"],["100000","2400000","12960000","27648000","25920000","8957952"]],"row_sums_preview":["2356","81928","2603920","77985952","2238550336","62235260032","1687546491136","44848554826240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-29d7c5a1e19b9d74","steps":["(\u03b8+1)\u00b2"],"edges":["e-e37593fec6045406"],"text":"A038314 \u2192 (\u03b8+1)\u00b2"},"headline":"A038314 \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:             400,              3840,               5184;\n  1:            8000,            115200,             311040,             221184;\n  2:          160000,           3072000,           12441600,           17694720,             8294400;\n  3:         3200000,          76800000,          414720000,          884736000,           829440000,           286654464;\n  4:        64000000,        1843200000,        12441600000,        35389440000,         49766400000,         34398535680,           9364045824;\n  5:      1280000000,       43008000000,       348364800000,      1238630400000,       2322432000000,       2407897497600,        1310966415360,         293534171136;\n  6:     25600000000,      983040000000,      9289728000000,     39636172800000,      92897280000000,     128421199872000,      104877313228800,       46965467381760,        8916100448256;\n  7:    512000000000,    22118400000000,    238878720000000,   1189085184000000,    3344302080000000,    5778953994240000,     6292638793728000,     4226892064358400,     1604898080686080,     264180754022400;\n  8:  10240000000000,   491520000000000,   5971968000000000,  33973862400000000,  111476736000000000,  231158159769600000,   314631939686400000,   281792804290560000,   160489808068608000,   52836150804480000,    7671809096810496;\n  9: 204800000000000, 10813440000000000, 145981440000000000, 934281216000000000, 3503554560000000000, 8475799191552000000, 13843805346201600000, 15498604235980800000, 11769252591697920000, 5811976588492800000, 1687798001298309120, 219122084616339456;\n","recurrence_sentence":"The row polynomials satisfy (25+(90*n+150)*x+(36*n^2+144*n+144)*x^2)*P(n,x) = (500+(1800*n+5400)*x+(720*n^2+6480*n+12240)*x^2+(864*n^2+5184*n+7776)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 3840*x + 5184*x^2.","mathematica":"P[0, x_] := 400 + 3840*x + 5184*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((500 + (1800*n + 5400)*x + (720*n^2 + 6480*n + 12240)*x^2 + (864*n^2 + 5184*n + 7776)*x^3)*P[n - 1, x])/(25 + (90*n + 150)*x + (36*n^2 + 144*n + 144)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 400 + 3840*x + 5184*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((500 + (1800*n + 5400)*x + (720*n^2 + 6480*n + 12240)*x^2 + (864*n^2 + 5184*n + 7776)*x^3)*v[i - 1])/(25 + (90*n + 150)*x + (36*n^2 + 144*n + 144)*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-6874f381e9c7e0a2","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-889e30cc3b49f76a","e-8784f281433f7aa1"],"text":"A038334 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A038334 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}