{"id":"n-62834ee6c7f6494c","revision":0,"first_index":0,"fingerprint":"f6-7fe862d483d8a327","name":"theta_plus_one_squared of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["225","360","81"],["3375","8100","3645","432"],["50625","162000","109350","25920","2025"],["759375","3037500","2733750","972000","151875","8748"],["11390625","54675000","61509375","29160000","6834375","787320","35721"],["170859375","956812500","1291696875","765450000","239203125","41334300","3750705","139968"],["2562890625","16402500000","25833937500","18370800000","7176093750","1653372000","225042300","16796160","531441"],["38443359375","276792187500","498225937500","413343000000","193754531250","55801305000","10126903500","1133740800","71744535","1968300"],["576650390625","4613203125000","9341736328125","8857350000000","4843863281250","1674039150000","379758881250","56687040000","5380840125","295245000","7144929"],["8649755859375","76117851562500","171265166015625","182682843750000","114176777343750","46036076625000","12532043081250","2338340400000","295946206875","24357712500","1178913285","25509168"],["129746337890625","1245564843750000","3082772988281250","3653656875000000","2568977490234375","1183784827500000","375961292437500","84180254400000","13317579309375","1461462750000","106102195650","4591650240","89813529"],["1946195068359375","20240428710937500","54649157519531250","71246309062500000","55661178955078125","28854755170312500","10473207432187500","2735858268000000","519385593065625","71246309062500","6896642717250","447685898400","17513638155","312487308"],["29192926025390625","326960771484375000","956360256591796875","1360156809375000000","1168884758056640625","673277620640625000","274921695094921875","82075748040000000","18178495757296875","2992344980625000","362073742655625","31338012888000","1838932006275","65622334680","1076168025"],["437893890380859375","5254726684570312500","16552389056396484375","25502940175781250000","23909006414794921875","15148746464414062500","6873042377373046875","2308380413625000000","584308792198828125","112212936773437500","16293318419503125","1762763224950000","137919900470625","7382512651500","242137805625","3673320192"],["6568408355712890625","84075626953125000000","283755240966796875000","470823510937500000000","478180128295898437500","330518104678125000000","164953017056953125000","61556811030000000000","17529263765964843750","3847300689375000000","651732736780125000","84612634797600000","8275194028237500","590601012120000","29056536675000","881596846080","12440502369"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["666","15552","349920","7663248","164392416","3469246848","72241963776","1487694677760","30348971426304","614120362979328","12339534735212544","246405084243775488","4893396743432724480","96701848836176904192","1902552653846643277824"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["225","360","81"]],"recurrence":{"denominator":{"coeffs":[["25","25","4"],["0","15","4"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["375","675","255","27"],["0","225","135","18"],["0","0","15","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193075,"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-7ca0a00eac64115a"]},{"id":193076,"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-7ca0a00eac64115a"]},{"id":193077,"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-7ca0a00eac64115a"]},{"id":193078,"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-7ca0a00eac64115a"]}],"region_exact":{},"incoming_edges":[{"id":"e-7ca0a00eac64115a","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-62834ee6c7f6494c","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":"4566035876e63199ad2236b90a1e0aae1075a36f2f7b375aa796f239cd790392","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-8447c9ce68adee7a","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","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-1de7e3ce7eb9449b","kind":"fingerprint_related","node_a_id":"n-ed584f5c1991eac3","node_b_id":"n-62834ee6c7f6494c","fingerprint":"f6-7fe862d483d8a327","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7fe862d483d8a327"},"other_id":"n-ed584f5c1991eac3","other":{"id":"n-ed584f5c1991eac3","name":"theta_plus_one_squared of basis_map_A013610 of Triangle of coefficients in expansion of (14 + x)^n.","first_index":0,"fingerprint":"f6-7fe862d483d8a327","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["15","12"],["225","360","81"]],"card_rows":[["1"],["15","12"],["225","360","81"],["3375","8100","3645","432"]],"row_sums_preview":["1","27","666","15552","349920","7663248","164392416","3469246848"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a679863398df5985","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-011a72a323f5e83e","e-85a6544982643a4c"],"text":"A147716 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A147716 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-48fd8f8722d20171","kind":"fingerprint_related","node_a_id":"n-09df7bfcd556014d","node_b_id":"n-62834ee6c7f6494c","fingerprint":"f6-7fe862d483d8a327","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7fe862d483d8a327"},"other_id":"n-09df7bfcd556014d","other":{"id":"n-09df7bfcd556014d","name":"theta_plus_one_squared of Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).","first_index":0,"fingerprint":"f6-7fe862d483d8a327","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5","4"],["25","40","9"]],"card_rows":[["1"],["5","4"],["25","40","9"],["125","300","135","16"]],"row_sums_preview":["1","9","74","576","4320","31536","225504","1586304"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b91a7a037fb3513","steps":["(\u03b8+1)\u00b2"],"edges":["e-70ae4e02dc5e9d42"],"text":"A038243 \u2192 (\u03b8+1)\u00b2"},"headline":"A038243 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-9791f36b31a54baa","kind":"fingerprint_related","node_a_id":"n-c4b5c836496940ed","node_b_id":"n-62834ee6c7f6494c","fingerprint":"f6-7fe862d483d8a327","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7fe862d483d8a327"},"other_id":"n-c4b5c836496940ed","other":{"id":"n-c4b5c836496940ed","name":"theta_plus_one_squared of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-7fe862d483d8a327","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["10","8"],["100","160","36"]],"card_rows":[["1"],["10","8"],["100","160","36"],["1000","2400","1080","128"]],"row_sums_preview":["1","18","296","4608","69120","1009152","14432256","203046912"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-88ca34e9877f4f47","e-e7029282b4d283cc"],"text":"A038291 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A038291 \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:           225,            360,              81;\n  1:          3375,           8100,            3645,             432;\n  2:         50625,         162000,          109350,           25920,            2025;\n  3:        759375,        3037500,         2733750,          972000,          151875,           8748;\n  4:      11390625,       54675000,        61509375,        29160000,         6834375,         787320,          35721;\n  5:     170859375,      956812500,      1291696875,       765450000,       239203125,       41334300,        3750705,        139968;\n  6:    2562890625,    16402500000,     25833937500,     18370800000,      7176093750,     1653372000,      225042300,      16796160,       531441;\n  7:   38443359375,   276792187500,    498225937500,    413343000000,    193754531250,    55801305000,    10126903500,    1133740800,     71744535,     1968300;\n  8:  576650390625,  4613203125000,   9341736328125,   8857350000000,   4843863281250,  1674039150000,   379758881250,   56687040000,   5380840125,   295245000,    7144929;\n  9: 8649755859375, 76117851562500, 171265166015625, 182682843750000, 114176777343750, 46036076625000, 12532043081250, 2338340400000, 295946206875, 24357712500, 1178913285, 25509168;\n","recurrence_sentence":"The row polynomials satisfy (25+(15*n+25)*x+(n^2+4*n+4)*x^2)*P(n,x) = (375+(225*n+675)*x+(15*n^2+135*n+255)*x^2+(3*n^2+18*n+27)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 360*x + 81*x^2.","mathematica":"P[0, x_] := 225 + 360*x + 81*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((375 + (225*n + 675)*x + (15*n^2 + 135*n + 255)*x^2 + (3*n^2 + 18*n + 27)*x^3)*P[n - 1, x])/(25 + (15*n + 25)*x + (n^2 + 4*n + 4)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 225 + 360*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((375 + (225*n + 675)*x + (15*n^2 + 135*n + 255)*x^2 + (3*n^2 + 18*n + 27)*x^3)*v[i - 1])/(25 + (15*n + 25)*x + (n^2 + 4*n + 4)*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-615df6ea54d3bbef","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-5083e5230277d126","e-7ca0a00eac64115a"],"text":"A038329 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A038329 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}
