{"id":"n-272e88c4cfcd9298","revision":0,"first_index":0,"fingerprint":"f6-dc876aec23f20348","name":"theta_plus_one_squared of basis_map_A013609 of Triangle of coefficients in expansion of (2+3x)^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"],["5","24"],["25","240","324"],["125","1800","4860","3456"],["625","12000","48600","69120","32400"],["3125","75000","405000","864000","810000","279936"],["15625","450000","3037500","8640000","12150000","8398080","2286144"],["78125","2625000","21262500","75600000","141750000","146966400","80015040","17915904"],["390625","15000000","141750000","604800000","1417500000","1959552000","1600300800","716636160","136048896"],["1953125","84375000","911250000","4536000000","12757500000","22044960000","24004512000","16124313600","6122200320","1007769600"],["9765625","468750000","5695312500","32400000000","106312500000","220449600000","300056400000","268738560000","153055008000","50388480000","7316407296"],["48828125","2578125000","34804687500","222750000000","835312500000","2020788000000","3300620400000","3695155200000","2806008480000","1385683200000","402402401280","52242776064"],["244140625","14062500000","208828125000","1485000000000","6264843750000","17321040000000","33006204000000","44341862400000","42090127200000","27713664000000","12072072038400","3134566563840","367876214784"],["1220703125","76171875000","1233984375000","9652500000000","45246093750000","140733450000000","306486180000000","480370176000000","547171653600000","450347040000000","261561560832000","101873413324800","23911953960960","2559896027136"],["6103515625","410156250000","7198242187500","61425000000000","316722656250000","1094593500000000","2681754075000000","4803701760000000","6383669292000000","6304858560000000","4577327314560000","2377046310912000","836918388633600","179192721899520","17631936921600"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","29","589","10241","162745","2437061","34977349","486212969","6591978481","87594833645","1144890783421","14758394597969","188020390932649","2371225294448021","29642456018129845"],"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":[["25","-30","0"],["0","90","0"],["0","0","36"]]},"terms":[{"coeff":{"coeffs":[["125","450","540","216"],["0","450","900","432"],["0","0","180","216"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":186142,"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-543e3e49cb8c0843"]},{"id":186143,"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-543e3e49cb8c0843"]},{"id":186144,"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-543e3e49cb8c0843"]},{"id":186145,"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-543e3e49cb8c0843"]}],"region_exact":{},"incoming_edges":[{"id":"e-543e3e49cb8c0843","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-272e88c4cfcd9298","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":"fc4cf44475dd426e2915e72cdbc696794e43b08a995ae647079bd6ba25440379","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-c1936dc488503f26","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients in expansion of (2+3x)^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-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-1fffa59090d6ab2a","other":{"id":"n-1fffa59090d6ab2a","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.","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":[["400","3840","5184"],["8000","115200","311040","221184"],["160000","3072000","12441600","17694720","8294400"]],"card_rows":[["400","3840","5184"],["8000","115200","311040","221184"],["160000","3072000","12441600","17694720","8294400"],["3200000","76800000","414720000","884736000","829440000","286654464"]],"row_sums_preview":["9424","655424","41662720","2495550464","143267221504","7966113284096","432011901730816","22962460071034880"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"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}},{"id":"l-f25e59d03514c103","kind":"fingerprint_related","node_a_id":"n-272e88c4cfcd9298","node_b_id":"n-d6ec43e2b2ecc097","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:       1;\n  1:       5,       24;\n  2:      25,      240,       324;\n  3:     125,     1800,      4860,       3456;\n  4:     625,    12000,     48600,      69120,       32400;\n  5:    3125,    75000,    405000,     864000,      810000,      279936;\n  6:   15625,   450000,   3037500,    8640000,    12150000,     8398080,     2286144;\n  7:   78125,  2625000,  21262500,   75600000,   141750000,   146966400,    80015040,    17915904;\n  8:  390625, 15000000, 141750000,  604800000,  1417500000,  1959552000,  1600300800,   716636160,  136048896;\n  9: 1953125, 84375000, 911250000, 4536000000, 12757500000, 22044960000, 24004512000, 16124313600, 6122200320, 1007769600;\n","recurrence_sentence":"The row polynomials satisfy (25+(90*n-30)*x+36*n^2*x^2)*P(n,x) = (125+(450*n+450)*x+(180*n^2+900*n+540)*x^2+(216*n^2+432*n+216)*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[((125 + (450*n + 450)*x + (180*n^2 + 900*n + 540)*x^2 + (216*n^2 + 432*n + 216)*x^3)*P[n - 1, x])/(25 + (90*n - 30)*x + 36*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] = ((125 + (450*n + 450)*x + (180*n^2 + 900*n + 540)*x^2 + (216*n^2 + 432*n + 216)*x^3)*v[i - 1])/(25 + (90*n - 30)*x + 36*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-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}