{"id":"n-f482e744be1cefd4","revision":0,"first_index":0,"fingerprint":"f6-6ae5c48b72ea3401","name":"theta_plus_one of Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).","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":true,"rows":[["1"],["5","2"],["25","20","3"],["125","150","45","4"],["625","1000","450","80","5"],["3125","6250","3750","1000","125","6"],["15625","37500","28125","10000","1875","180","7"],["78125","218750","196875","87500","21875","3150","245","8"],["390625","1250000","1312500","700000","218750","42000","4900","320","9"],["1953125","7031250","8437500","5250000","1968750","472500","73500","7200","405","10"],["9765625","39062500","52734375","37500000","16406250","4725000","918750","120000","10125","500","11"],["48828125","214843750","322265625","257812500","128906250","43312500","10106250","1650000","185625","13750","605","12"],["244140625","1171875000","1933593750","1718750000","966796875","371250000","101062500","19800000","2784375","275000","18150","720","13"],["1220703125","6347656250","11425781250","11171875000","6982421875","3016406250","938437500","214500000","36196875","4468750","393250","23400","845","14"],["6103515625","34179687500","66650390625","71093750000","48876953125","23460937500","8211328125","2145000000","422296875","62562500","6881875","546000","29575","980","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","48","324","2160","14256","93312","606528","3919104","25194240","161243136","1027924992","6530347008","41358864384","261213880320"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["5","0"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["25","10","1"],["0","5","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180765,"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-915fc3d2874d3b78"]},{"id":180766,"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-915fc3d2874d3b78"]},{"id":180767,"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-915fc3d2874d3b78"]},{"id":180768,"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-915fc3d2874d3b78"]}],"region_exact":{},"incoming_edges":[{"id":"e-915fc3d2874d3b78","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-f482e744be1cefd4","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":"d67a9a8e5fd95df7e32f34f0a10a864f8494acb938313842ba369cd4a2ce3a98","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-7b91a7a037fb3513","role":"sequence","name":"Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,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-0a2f11cfcdf0b7a8","kind":"fingerprint_related","node_a_id":"n-f482e744be1cefd4","node_b_id":"n-d4d842f5bb398cf3","fingerprint":"f6-6ae5c48b72ea3401","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6ae5c48b72ea3401"},"other_id":"n-d4d842f5bb398cf3","other":{"id":"n-d4d842f5bb398cf3","name":"theta_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-6ae5c48b72ea3401","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["225","180","27"],["3375","4050","1215","108"],["50625","81000","36450","6480","405"]],"card_rows":[["225","180","27"],["3375","4050","1215","108"],["50625","81000","36450","6480","405"],["759375","1518750","911250","243000","30375","1458"]],"row_sums_preview":["432","8748","174960","3464208","68024448","1326476736","25713241344","495898225920"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-5083e5230277d126","e-051c9abffcf73621"],"text":"A038329 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A038329 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}},{"id":"l-1faf718a74b742ac","kind":"fingerprint_related","node_a_id":"n-f482e744be1cefd4","node_b_id":"n-ececab726f2e6291","fingerprint":"f6-6ae5c48b72ea3401","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6ae5c48b72ea3401"},"other_id":"n-ececab726f2e6291","other":{"id":"n-ececab726f2e6291","name":"theta_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-6ae5c48b72ea3401","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","4"],["100","80","12"]],"card_rows":[["1"],["10","4"],["100","80","12"],["1000","1200","360","32"]],"row_sums_preview":["1","14","192","2592","34560","456192","5971968","77635584"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013609","\u03b8+1"],"edges":["e-88ca34e9877f4f47","e-fb893e59a50c4636"],"text":"A038291 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A038291 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}},{"id":"l-b9f0004e96a0977a","kind":"fingerprint_related","node_a_id":"n-f482e744be1cefd4","node_b_id":"n-90989d1b314934e8","fingerprint":"f6-6ae5c48b72ea3401","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6ae5c48b72ea3401"},"other_id":"n-90989d1b314934e8","other":{"id":"n-90989d1b314934e8","name":"theta_plus_one of basis_map_A013610 of Triangle of coefficients in expansion of (14 + x)^n.","first_index":0,"fingerprint":"f6-6ae5c48b72ea3401","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["15","6"],["225","180","27"]],"card_rows":[["1"],["15","6"],["225","180","27"],["3375","4050","1215","108"]],"row_sums_preview":["1","21","432","8748","174960","3464208","68024448","1326476736"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a679863398df5985","steps":["basis A013610","\u03b8+1"],"edges":["e-011a72a323f5e83e","e-0d5efd37416f29b9"],"text":"A147716 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A147716 \u2192 basis A013610 \u2192 \u03b8+1","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:       5,       2;\n  2:      25,      20,       3;\n  3:     125,     150,      45,       4;\n  4:     625,    1000,     450,      80,       5;\n  5:    3125,    6250,    3750,    1000,     125,      6;\n  6:   15625,   37500,   28125,   10000,    1875,    180,     7;\n  7:   78125,  218750,  196875,   87500,   21875,   3150,   245,    8;\n  8:  390625, 1250000, 1312500,  700000,  218750,  42000,  4900,  320,   9;\n  9: 1953125, 7031250, 8437500, 5250000, 1968750, 472500, 73500, 7200, 405, 10;\n","recurrence_sentence":"The row polynomials satisfy (5+n*x)*P(n,x) = (25+(5*n+10)*x+(n+1)*x^2)*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[((25 + (5*n + 10)*x + (n + 1)*x^2)*P[n - 1, x])/(5 + n*x)]];\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] = ((25 + (5*n + 10)*x + (n + 1)*x^2)*v[i - 1])/(5 + n*x)); 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-7b91a7a037fb3513","steps":["\u03b8+1"],"edges":["e-915fc3d2874d3b78"],"text":"A038243 \u2192 \u03b8+1"},"headline":"A038243 \u2192 \u03b8+1","chain":null}