{"id":"n-af0d0a00ea5542fb","revision":0,"first_index":0,"fingerprint":"f6-6fb0656952daab5a","name":"theta_plus_one_squared of derivative of Triangle of coefficients in expansion of (1+4x)^n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["4"],["8","128"],["12","384","1728"],["16","768","6912","16384"],["20","1280","17280","81920","128000"],["24","1920","34560","245760","768000","884736"],["28","2688","60480","573440","2688000","6193152","5619712"],["32","3584","96768","1146880","7168000","24772608","44957696","33554432"],["36","4608","145152","2064384","16128000","74317824","202309632","301989888","191102976"],["40","5760","207360","3440640","32256000","185794560","674365440","1509949440","1911029760","1048576000"],["44","7040","285120","5406720","59136000","408748032","1854504960","5536481280","10510663680","11534336000","5582618624"],["48","8448","380160","8110080","101376000","817496064","4450811904","16609443840","42042654720","69206016000","66991423488","28991029248"],["52","9984","494208","11714560","164736000","1518206976","9643425792","43184553984","136638627840","299892736000","435444252672","376883380224","147438174208"],["56","11648","628992","16400384","256256000","2656862208","19286851584","100763959296","382588157952","1049624576000","2032073179136","2638183661568","2064134438912","736586891264"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","4","136","2124","24080","228500","1935000","15137500","111700000","788062500","5365625000","35492187500","229218750000","1450820312500","9026171875000"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["4"]],"recurrence":{"denominator":{"coeffs":[["-1","16","-16"],["1","-28","48"],["0","12","-48"],["0","0","16"]]},"terms":[{"coeff":{"coeffs":[["0","0","0","0"],["1","0","-16","0"],["0","12","48","0"],["0","0","16","64"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180849,"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-b0c3ab126687eaf5"]},{"id":180850,"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-b0c3ab126687eaf5"]},{"id":180851,"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-b0c3ab126687eaf5"]}],"region_exact":{},"incoming_edges":[{"id":"e-b0c3ab126687eaf5","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-af0d0a00ea5542fb","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":"fcd7e98bdbdf13e4288c0881b25f5a75109160b70d5a345a9395b3a28e868a42","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-8d4f9cf28f37366a","role":"sequence","name":"derivative of Triangle of coefficients in expansion of (1+4x)^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-04fc587baeb89cbb","kind":"fingerprint_related","node_a_id":"n-af0d0a00ea5542fb","node_b_id":"n-428df32f2c9ad121","fingerprint":"f6-6fb0656952daab5a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6fb0656952daab5a"},"other_id":"n-428df32f2c9ad121","other":{"id":"n-428df32f2c9ad121","name":"theta_plus_one_squared of basis_map_A013611 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-6fb0656952daab5a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1","32","144"],["1","48","432","1024"],["1","64","864","4096","6400"]],"card_rows":[["1","32","144"],["1","48","432","1024"],["1","64","864","4096","6400"],["1","80","1440","10240","32000","36864"]],"row_sums_preview":["177","1505","11425","80625","540625","3490625","21890625","134140625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-b91bf4c52fd06745","e-e9892f449c7e752d"],"text":"basis-map coefficients of A112227 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A112227 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-1eeaefa01f8e3902","kind":"fingerprint_related","node_a_id":"n-af0d0a00ea5542fb","node_b_id":"n-fead27c2a1032954","fingerprint":"f6-6fb0656952daab5a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6fb0656952daab5a"},"other_id":"n-fead27c2a1032954","other":{"id":"n-fead27c2a1032954","name":"theta_plus_one_squared of basis_map_A013611 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-6fb0656952daab5a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","16"],["2","64","288"]],"card_rows":[["1"],["1","16"],["2","64","288"],["6","288","2592","6144"]],"row_sums_preview":["1","17","354","9030","274200","9675000","389250000","17592750000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-3656191e13afb7f3","e-ec8546468386dc24"],"text":"A136572 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A136572 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-7a69f34894ea956f","kind":"fingerprint_related","node_a_id":"n-c8692c346ff4c403","node_b_id":"n-af0d0a00ea5542fb","fingerprint":"f6-6fb0656952daab5a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6fb0656952daab5a"},"other_id":"n-c8692c346ff4c403","other":{"id":"n-c8692c346ff4c403","name":"theta_plus_one_squared of basis_map_A013611 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-6fb0656952daab5a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","16"],["1","32","144"]],"card_rows":[["1"],["1","16"],["1","32","144"],["1","48","432","1024"]],"row_sums_preview":["1","17","177","1505","11425","80625","540625","3490625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-f93ad378f9476e41","e-d257369cc4f911ca"],"text":"basis-map coefficients of A026729 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A026729 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-d61d074dce3d17a6","kind":"fingerprint_related","node_a_id":"n-af0d0a00ea5542fb","node_b_id":"n-cb27801678970979","fingerprint":"f6-6fb0656952daab5a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6fb0656952daab5a"},"other_id":"n-cb27801678970979","other":{"id":"n-cb27801678970979","name":"theta_plus_one_squared of basis_map_A013611 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-6fb0656952daab5a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","32"],["4","128","576"]],"card_rows":[["1"],["2","32"],["4","128","576"],["8","384","3456","8192"]],"row_sums_preview":["1","34","708","12040","182800","2580000","34600000","446800000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-d7d2422771947afc","e-bdca32644e8032ee"],"text":"A130123 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A130123 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-f5bb4de44652fa94","kind":"fingerprint_related","node_a_id":"n-16532974d073ecae","node_b_id":"n-af0d0a00ea5542fb","fingerprint":"f6-6fb0656952daab5a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6fb0656952daab5a"},"other_id":"n-16532974d073ecae","other":{"id":"n-16532974d073ecae","name":"theta_plus_one_squared of Triangle of coefficients in expansion of (1+4x)^n.","first_index":0,"fingerprint":"f6-6fb0656952daab5a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","16"],["1","32","144"]],"card_rows":[["1"],["1","16"],["1","32","144"],["1","48","432","1024"]],"row_sums_preview":["1","17","177","1505","11425","80625","540625","3490625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2e0409c0014ec240","steps":["(\u03b8+1)\u00b2"],"edges":["e-738497c1b246462b"],"text":"A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A013611 \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:  0;\n  1:  4;\n  2:  8,  128;\n  3: 12,  384,   1728;\n  4: 16,  768,   6912,   16384;\n  5: 20, 1280,  17280,   81920,   128000;\n  6: 24, 1920,  34560,  245760,   768000,   884736;\n  7: 28, 2688,  60480,  573440,  2688000,  6193152,   5619712;\n  8: 32, 3584,  96768, 1146880,  7168000, 24772608,  44957696,  33554432;\n  9: 36, 4608, 145152, 2064384, 16128000, 74317824, 202309632, 301989888, 191102976;\n","recurrence_sentence":"The row polynomials satisfy (n-1+(12*n^2-28*n+16)*x+(16*n^3-48*n^2+48*n-16)*x^2)*P(n,x) = (n+12*n^2*x+(16*n^3+48*n^2-16*n)*x^2+64*n^3*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((n + 12*n^2*x + (16*n^3 + 48*n^2 - 16*n)*x^2 + 64*n^3*x^3)*P[n - 1, x])/(n - 1 + (12*n^2 - 28*n + 16)*x + (16*n^3 - 48*n^2 + 48*n - 16)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((n + 12*n^2*x + (16*n^3 + 48*n^2 - 16*n)*x^2 + 64*n^3*x^3)*v[i - 1])/(n - 1 + (12*n^2 - 28*n + 16)*x + (16*n^3 - 48*n^2 + 48*n - 16)*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-2e0409c0014ec240","steps":["D","(\u03b8+1)\u00b2"],"edges":["e-b04b02754d5cebbf","e-b0c3ab126687eaf5"],"text":"A013611 \u2192 D \u2192 (\u03b8+1)\u00b2"},"headline":"A013611 \u2192 D \u2192 (\u03b8+1)\u00b2","chain":null}