{"id":"n-5f871260d2ebe306","revision":0,"first_index":0,"fingerprint":"f6-e4d4f060223d7e66","name":"theta_plus_one_squared of basis_map_A013611 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","16"],["225","480","144"],["3375","10800","6480","1024"],["50625","216000","194400","61440","6400"],["759375","4050000","4860000","2304000","480000","36864"],["11390625","72900000","109350000","69120000","21600000","3317760","200704"],["170859375","1275750000","2296350000","1814400000","756000000","174182400","21073920","1048576"],["2562890625","21870000000","45927000000","43545600000","22680000000","6967296000","1264435200","125829120","5308416"],["38443359375","369056250000","885735000000","979776000000","612360000000","235146240000","56899584000","8493465600","716636160","26214400"],["576650390625","6150937500000","16607531250000","20995200000000","15309000000000","7054387200000","2133734400000","424673280000","53747712000","3932160000","126877696"],["8649755859375","101490468750000","304471406250000","433026000000000","360855000000000","193995648000000","70413235200000","17517772800000","2956124160000","324403200000","20934819840","603979776"],["129746337890625","1660753125000000","5480485312500000","8660520000000000","8119237500000000","4988459520000000","2112397056000000","630639820800000","133025587200000","19464192000000","1884133785600","108716359680","2835349504"],["1946195068359375","26987238281250000","97154057812500000","168880140000000000","175916812500000000","121593700800000000","58845346560000000","20495794176000000","5187997900800000","948879360000000","122468696064000","10599845068800","552893153280","13153337344"],["29192926025390625","435947695312500000","1700196011718750000","3224075400000000000","3694253062500000000","2837186352000000000","1544690347200000000","614873825280000000","181579926528000000","39852933120000000","6429606543360000","741989154816000","58053781094400","2762200842240","60397977600"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","31","849","21679","528865","12490239","287879089","6509664271","144948359361","3186652749535","69309920770321","1493721353018991","31936724136885409","678089797046532799","14309080951762730865"],"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","-60","0"],["0","180","0"],["0","0","16"]]},"terms":[{"coeff":{"coeffs":[["3375","2700","720","64"],["0","2700","1200","128"],["0","0","240","64"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190478,"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-067247ed8daf773c"]},{"id":190479,"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-067247ed8daf773c"]},{"id":190480,"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-067247ed8daf773c"]},{"id":190481,"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-067247ed8daf773c"]}],"region_exact":{},"incoming_edges":[{"id":"e-067247ed8daf773c","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-5f871260d2ebe306","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":"6b9385ca1d55745cf901f220794861d3e88e302a47529c513d4ec3c10b288017","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-60783defa9dbe6a9","role":"sequence","name":"basis_map_A013611 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,           16;\n  2:         225,          480,          144;\n  3:        3375,        10800,         6480,         1024;\n  4:       50625,       216000,       194400,        61440,         6400;\n  5:      759375,      4050000,      4860000,      2304000,       480000,        36864;\n  6:    11390625,     72900000,    109350000,     69120000,     21600000,      3317760,      200704;\n  7:   170859375,   1275750000,   2296350000,   1814400000,    756000000,    174182400,    21073920,    1048576;\n  8:  2562890625,  21870000000,  45927000000,  43545600000,  22680000000,   6967296000,  1264435200,  125829120,   5308416;\n  9: 38443359375, 369056250000, 885735000000, 979776000000, 612360000000, 235146240000, 56899584000, 8493465600, 716636160, 26214400;\n","recurrence_sentence":"The row polynomials satisfy (225+(180*n-60)*x+16*n^2*x^2)*P(n,x) = (3375+(2700*n+2700)*x+(240*n^2+1200*n+720)*x^2+(64*n^2+128*n+64)*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 + (2700*n + 2700)*x + (240*n^2 + 1200*n + 720)*x^2 + (64*n^2 + 128*n + 64)*x^3)*P[n - 1, x])/(225 + (180*n - 60)*x + 16*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 + (2700*n + 2700)*x + (240*n^2 + 1200*n + 720)*x^2 + (64*n^2 + 128*n + 64)*x^3)*v[i - 1])/(225 + (180*n - 60)*x + 16*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 A013611","(\u03b8+1)\u00b2"],"edges":["e-8d978ffb07d0025e","e-067247ed8daf773c"],"text":"A147716 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A147716 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}