{"id":"n-b7e42fe8311c70bc","revision":0,"first_index":0,"fingerprint":"f6-2ead7e86fdf601f9","name":"theta_plus_one_squared of basis_map_A013610 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","60"],["64","960","2025"],["512","11520","48600","54000"],["4096","122880","777600","1728000","1265625"],["32768","1228800","10368000","34560000","50625000","27337500"],["262144","11796480","124416000","552960000","1215000000","1312200000","558140625"],["2097152","110100480","1393459200","7741440000","22680000000","36741600000","31255875000","10935000000"],["16777216","1006632960","14863564800","99090432000","362880000000","783820800000","1000188000000","699840000000","207594140625"],["134217728","9059696640","152882380800","1189085184000","5225472000000","14108774400000","24004512000000","25194240000000","14946778125000","3844335937500"],["1073741824","80530636800","1528823808000","13589544960000","69672960000000","225740390400000","480090240000000","671846400000000","597871125000000","307546875000000","69774697265625"],["8589934592","708669603840","14948499456000","149484994560000","875888640000000","3310859059200000","8449588224000000","14780620800000000","17537553000000000","13532062500000000","6140173359375000","1245564843750000"],["68719476736","6184752906240","143505594777600","1594506608640000","10510663680000000","45406067097600000","135193411584000000","283787919360000000","420901272000000000","433026000000000000","294728321250000000","119574225000000000","21927131103515625"],["549755813888","53601191854080","1356780168806400","16582868729856000","121456558080000000","590278872268800000","2008587829248000000","4918990602240000000","8754746457600000000","11258676000000000000","10217248470000000000","6217859700000000000","2280421634765625000","381454233398437500"],["4398046511104","461794883665920","12663281575526400","168843754340352000","1360313450496000000","7345692632678400000","28120229609472000000","78703849635840000000","163421933875200000000","252194342400000000000","286082957160000000000","232133428800000000000","127703611546875000000","42722874140625000000","6568408355712890625"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","68","3049","114632","3898201","124152068","3774775249","110859571832","3169300347601","88675273941668","2437742660812249","66037461179879432","1766799276750916201","46767714157447192868","1226539614835745346049"],"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","-120","0"],["0","360","0"],["0","0","225"]]},"terms":[{"coeff":{"coeffs":[["512","2880","5400","3375"],["0","2880","9000","6750"],["0","0","1800","3375"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191460,"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-c6fc3e818b728cdc"]},{"id":191461,"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-c6fc3e818b728cdc"]},{"id":191462,"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-c6fc3e818b728cdc"]},{"id":191463,"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-c6fc3e818b728cdc"]}],"region_exact":{},"incoming_edges":[{"id":"e-c6fc3e818b728cdc","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-b7e42fe8311c70bc","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":"c2aac82767defea5395328a17a2aa2862eefcb800d4028b802f9671a3f5ad0e9","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-a6cd3f3657ee08fd","role":"sequence","name":"basis_map_A013610 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":[],"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,         60;\n  2:        64,        960,         2025;\n  3:       512,      11520,        48600,         54000;\n  4:      4096,     122880,       777600,       1728000,       1265625;\n  5:     32768,    1228800,     10368000,      34560000,      50625000,       27337500;\n  6:    262144,   11796480,    124416000,     552960000,    1215000000,     1312200000,      558140625;\n  7:   2097152,  110100480,   1393459200,    7741440000,   22680000000,    36741600000,    31255875000,    10935000000;\n  8:  16777216, 1006632960,  14863564800,   99090432000,  362880000000,   783820800000,  1000188000000,   699840000000,   207594140625;\n  9: 134217728, 9059696640, 152882380800, 1189085184000, 5225472000000, 14108774400000, 24004512000000, 25194240000000, 14946778125000, 3844335937500;\n","recurrence_sentence":"The row polynomials satisfy (64+(360*n-120)*x+225*n^2*x^2)*P(n,x) = (512+(2880*n+2880)*x+(1800*n^2+9000*n+5400)*x^2+(3375*n^2+6750*n+3375)*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 + (2880*n + 2880)*x + (1800*n^2 + 9000*n + 5400)*x^2 + (3375*n^2 + 6750*n + 3375)*x^3)*P[n - 1, x])/(64 + (360*n - 120)*x + 225*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 + (2880*n + 2880)*x + (1800*n^2 + 9000*n + 5400)*x^2 + (3375*n^2 + 6750*n + 3375)*x^3)*v[i - 1])/(64 + (360*n - 120)*x + 225*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 A013610","(\u03b8+1)\u00b2"],"edges":["e-c78a98c6ca91439c","e-c6fc3e818b728cdc"],"text":"A013622 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A013622 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}