{"id":"n-15ae131e7f255640","revision":0,"first_index":0,"fingerprint":"f6-472634f5a19ab425","name":"theta_plus_one_squared of basis_map_A007318 of Triangle of coefficients in expansion of (3+5x)^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"],["8","20"],["64","320","225"],["512","3840","5400","2000"],["4096","40960","86400","64000","15625"],["32768","409600","1152000","1280000","625000","112500"],["262144","3932160","13824000","20480000","15000000","5400000","765625"],["2097152","36700160","154828800","286720000","280000000","151200000","42875000","5000000"],["16777216","335544320","1651507200","3670016000","4480000000","3225600000","1372000000","320000000","31640625"],["134217728","3019898880","16986931200","44040192000","64512000000","58060800000","32928000000","11520000000","2278125000","195312500"],["1073741824","26843545600","169869312000","503316480000","860160000000","928972800000","658560000000","307200000000","91125000000","15625000000","1181640625"],["8589934592","236223201280","1660944384000","5536481280000","10813440000000","13624934400000","11590656000000","6758400000000","2673000000000","687500000000","103984375000","7031250000"],["68719476736","2061584302080","15945066086400","59055800320000","129761280000000","186856243200000","185450496000000","129761280000000","64152000000000","22000000000000","4991250000000","675000000000","41259765625"],["549755813888","17867063951360","150753352089600","614180323328000","1499463680000000","2429131161600000","2755264512000000","2249195520000000","1334361600000000","572000000000000","173030000000000","35100000000000","4291015625000","239257812500"],["4398046511104","153931627888640","1407031286169600","6253472382976000","16793993216000000","30229187788800000","38573703168000000","35987128320000000","24908083200000000","12812800000000000","4844840000000000","1310400000000000","240296875000000","26796875000000","1373291015625"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","28","609","11752","211081","3611868","59663929","959421112","15103085361","233675477308","3563927520049","53701184824872","800819979150841","11835427242220348","173547436077360969"],"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":[["64","-40","0"],["0","120","0"],["0","0","25"]]},"terms":[{"coeff":{"coeffs":[["512","960","600","125"],["0","960","1000","250"],["0","0","200","125"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187300,"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-0a10f3a658693f64"]},{"id":187301,"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-0a10f3a658693f64"]},{"id":187302,"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-0a10f3a658693f64"]},{"id":187303,"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-0a10f3a658693f64"]}],"region_exact":{},"incoming_edges":[{"id":"e-0a10f3a658693f64","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-15ae131e7f255640","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":"b7e062b2b8a403815cea3862d0270d789f85f9a1dd57dcbb131c3682985b1a68","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-93086473ff31e9a0","role":"sequence","name":"basis_map_A007318 of Triangle of coefficients in expansion of (3+5x)^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-8837c561c68f3a16","kind":"fingerprint_related","node_a_id":"n-15ae131e7f255640","node_b_id":"n-31c6ec19afe4c4d5","fingerprint":"f6-472634f5a19ab425","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-472634f5a19ab425"},"other_id":"n-31c6ec19afe4c4d5","other":{"id":"n-31c6ec19afe4c4d5","name":"theta_plus_one_squared of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","first_index":0,"fingerprint":"f6-472634f5a19ab425","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["256","1280","900"],["4096","30720","43200","16000"],["65536","655360","1382400","1024000","250000"]],"card_rows":[["256","1280","900"],["4096","30720","43200","16000"],["65536","655360","1382400","1024000","250000"],["1048576","13107200","36864000","40960000","20000000","3600000"]],"row_sums_preview":["2436","94016","3377296","115579776","3818491456","122805902336","3866389852416","119641844381696"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2bc1923905556bf6","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-772e88d8a50e6734","e-7f2d439081995f3e"],"text":"A038319 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A038319 \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:         8,         20;\n  2:        64,        320,         225;\n  3:       512,       3840,        5400,        2000;\n  4:      4096,      40960,       86400,       64000,       15625;\n  5:     32768,     409600,     1152000,     1280000,      625000,      112500;\n  6:    262144,    3932160,    13824000,    20480000,    15000000,     5400000,      765625;\n  7:   2097152,   36700160,   154828800,   286720000,   280000000,   151200000,    42875000,     5000000;\n  8:  16777216,  335544320,  1651507200,  3670016000,  4480000000,  3225600000,  1372000000,   320000000,   31640625;\n  9: 134217728, 3019898880, 16986931200, 44040192000, 64512000000, 58060800000, 32928000000, 11520000000, 2278125000, 195312500;\n","recurrence_sentence":"The row polynomials satisfy (64+(120*n-40)*x+25*n^2*x^2)*P(n,x) = (512+(960*n+960)*x+(200*n^2+1000*n+600)*x^2+(125*n^2+250*n+125)*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[((512 + (960*n + 960)*x + (200*n^2 + 1000*n + 600)*x^2 + (125*n^2 + 250*n + 125)*x^3)*P[n - 1, x])/(64 + (120*n - 40)*x + 25*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] = ((512 + (960*n + 960)*x + (200*n^2 + 1000*n + 600)*x^2 + (125*n^2 + 250*n + 125)*x^3)*v[i - 1])/(64 + (120*n - 40)*x + 25*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-c44f7114310405ad","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-31008ec5d02c6da5","e-0a10f3a658693f64"],"text":"A013622 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A013622 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}