{"id":"n-28612eb32b3440a9","revision":0,"first_index":0,"fingerprint":"f6-2751883146f8224a","name":"theta_plus_one_squared of reversal of Triangle read by rows: T(n,k) = number of vertices of degree k in an origami flip graph OFG(A2n).","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":[["8"],["18","48"],["32","256","144"],["50","800","1350","320"],["72","1920","6480","4608","600"],["98","3920","22050","31360","12250","1008"],["128","7168","60480","143360","112000","27648","1568"],["162","12096","142884","508032","661500","326592","55566","2304"],["200","19200","302400","1505280","2940000","2419200","823200","102400","3240"],["242","29040","588060","3902976","10672200","13172544","7470540","1858560","176418","4400"],["288","42240","1069200","9123840","33264000","57480192","48898080","20275200","3849120","288000","5808"],["338","59488","1840410","19631040","92020500","212015232","252504252","157048320","49691070","7436000","449878","7488"],["392","81536","3027024","39463424","231231000","684972288","1087710624","947122176","449513064","112112000","13565552","677376","9464"],["450","109200","4791150","74954880","536786250","1987642800","4058104050","4711449600","3130537410","1171170000","236185950","23587200","988650","11760"],["512","143360","7338240","135675904","1165964800","5276823552","13466893440","20102184960","17809279488","9327718400","2821634816","469647360","39370240","1404928","14400"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["8","66","432","2520","13680","70686","352352","1709136","8115120","37874980","174295968","792704016","3569485920","15936319350","70624094400"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["8"]],"recurrence":{"denominator":{"coeffs":[["-48","12","3"],["-112","28","7"],["-80","20","5"],["-16","4","1"]]},"terms":[{"coeff":{"coeffs":[["-192","48","12","0","0"],["-256","-1280","384","160","6"],["-112","-164","599","224","29"],["-16","4","-559","76","9"],["0","0","-16","-196","-28"],["0","0","0","-168","-16"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-624","1188","-456","-228","-12"],["0","-256","760","-484","-380","-58"],["0","-32","-40","1166","-136","-18"],["0","0","0","200","408","56"],["0","0","0","0","336","32"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-176","388","-122","-84","-6"],["0","0","-48","468","-145","-240","-35"],["0","0","0","16","348","-320","-44"],["0","0","0","0","168","-152","-16"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177933,"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-c94c8eae24ae549c"]},{"id":177934,"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-c94c8eae24ae549c"]},{"id":177935,"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-c94c8eae24ae549c"]},{"id":177936,"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-c94c8eae24ae549c"]}],"region_exact":{},"incoming_edges":[{"id":"e-c94c8eae24ae549c","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-28612eb32b3440a9","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":"089ecafb04ada3f05b8bc60a7dd4c71f6a7f54a9d284d97f5fd7a95ddff7685f","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-4d976aa636e25300","role":"sequence","name":"reversal of Triangle read by rows: T(n,k) = number of vertices of degree k in an origami flip graph OFG(A2n).","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-203da24cc9de25db","kind":"fingerprint_related","node_a_id":"n-d9e6a4222d7d90c1","node_b_id":"n-28612eb32b3440a9","fingerprint":"f6-2751883146f8224a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2751883146f8224a"},"other_id":"n-d9e6a4222d7d90c1","other":{"id":"n-d9e6a4222d7d90c1","name":"theta_plus_one_squared of Triangle of Dyck paths counted by number of long interior inclines.","first_index":0,"fingerprint":"f6-2751883146f8224a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["3","8"],["4","32","18"]],"card_rows":[["2"],["3","8"],["4","32","18"],["5","80","135","32"]],"row_sums_preview":["2","11","54","252","1140","5049","22022","94952"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c963abfc0266d66c","steps":["(\u03b8+1)\u00b2"],"edges":["e-ade1c1d52bd3c716"],"text":"A108838 \u2192 (\u03b8+1)\u00b2"},"headline":"A108838 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-9a41b8c2434ed72b","kind":"fingerprint_related","node_a_id":"n-79678d67d09978c9","node_b_id":"n-28612eb32b3440a9","fingerprint":"f6-2751883146f8224a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2751883146f8224a"},"other_id":"n-79678d67d09978c9","other":{"id":"n-79678d67d09978c9","name":"theta_plus_one_squared of derivative of Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.","first_index":0,"fingerprint":"f6-2751883146f8224a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["3","8"]],"card_rows":[["1"],["3","8"],["6","48","27"],["10","160","270","64"]],"row_sums_preview":["0","1","11","81","504","2850","15147","77077"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c2eaf5212a4dec13","steps":["D","(\u03b8+1)\u00b2"],"edges":["e-1c13b2418122affc","e-1d8c865f1528a1cd"],"text":"A001263 \u2192 D \u2192 (\u03b8+1)\u00b2"},"headline":"A001263 \u2192 D \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-e7a6fb63ec0beedc","kind":"fingerprint_related","node_a_id":"n-3cfdab3970f8c02c","node_b_id":"n-28612eb32b3440a9","fingerprint":"f6-2751883146f8224a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2751883146f8224a"},"other_id":"n-3cfdab3970f8c02c","other":{"id":"n-3cfdab3970f8c02c","name":"theta_plus_one_squared of derivative of narayana","first_index":1,"fingerprint":"f6-2751883146f8224a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["3","8"]],"card_rows":[["1"],["3","8"],["6","48","27"],["10","160","270","64"]],"row_sums_preview":["0","1","11","81","504","2850","15147","77077"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2a05d8340c329b21","steps":["D","(\u03b8+1)\u00b2"],"edges":["e-18f6a04857337ebb","e-809f751074ebae28"],"text":"narayana \u2192 D \u2192 (\u03b8+1)\u00b2"},"headline":"narayana \u2192 D \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:   8;\n  1:  18,    48;\n  2:  32,   256,    144;\n  3:  50,   800,   1350,     320;\n  4:  72,  1920,   6480,    4608,      600;\n  5:  98,  3920,  22050,   31360,    12250,     1008;\n  6: 128,  7168,  60480,  143360,   112000,    27648,    1568;\n  7: 162, 12096, 142884,  508032,   661500,   326592,   55566,    2304;\n  8: 200, 19200, 302400, 1505280,  2940000,  2419200,  823200,  102400,   3240;\n  9: 242, 29040, 588060, 3902976, 10672200, 13172544, 7470540, 1858560, 176418, 4400;\n","recurrence_sentence":"The row polynomials satisfy -(16*n^3+80*n^2+112*n+48-(4*n^3+20*n^2+28*n+12)*x-(n^3+5*n^2+7*n+3)*x^2)*P(n,x) = -(16*n^3+112*n^2+256*n+192-(4*n^3-164*n^2-1280*n+48)*x+(16*n^4+559*n^3-599*n^2-384*n-12)*x^2+(168*n^5+196*n^4-76*n^3-224*n^2-160*n)*x^3+(16*n^5+28*n^4-9*n^3-29*n^2-6*n)*x^4)*P(n-1,x) - ((32*n^2+256*n+624)*x+(40*n^2-760*n-1188)*x^2-(200*n^3+1166*n^2-484*n-456)*x^3-(336*n^4+408*n^3-136*n^2-380*n-228)*x^4-(32*n^4+56*n^3-18*n^2-58*n-12)*x^5)*d/dx P(n-1,x) - ((48*n+176)*x^2-(16*n^2+468*n+388)*x^3-(168*n^3+348*n^2-145*n-122)*x^4+(152*n^3+320*n^2+240*n+84)*x^5+(16*n^3+44*n^2+35*n+6)*x^6)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 8.","mathematica":"P[0, x_] := 8;\nP[n_, x_] := P[n, x] = Expand[Together[(-(16*n^3 + 112*n^2 + 256*n + 192 - (4*n^3 - 164*n^2 - 1280*n + 48)*x + (16*n^4 + 559*n^3 - 599*n^2 - 384*n - 12)*x^2 + (168*n^5 + 196*n^4 - 76*n^3 - 224*n^2 - 160*n)*x^3 + (16*n^5 + 28*n^4 - 9*n^3 - 29*n^2 - 6*n)*x^4)*P[n - 1, x] - ((32*n^2 + 256*n + 624)*x + (40*n^2 - 760*n - 1188)*x^2 - (200*n^3 + 1166*n^2 - 484*n - 456)*x^3 - (336*n^4 + 408*n^3 - 136*n^2 - 380*n - 228)*x^4 - (32*n^4 + 56*n^3 - 18*n^2 - 58*n - 12)*x^5)*D[P[n - 1, x], x] - ((48*n + 176)*x^2 - (16*n^2 + 468*n + 388)*x^3 - (168*n^3 + 348*n^2 - 145*n - 122)*x^4 + (152*n^3 + 320*n^2 + 240*n + 84)*x^5 + (16*n^3 + 44*n^2 + 35*n + 6)*x^6)*D[P[n - 1, x], {x, 2}])/(-16*n^3 - 80*n^2 - 112*n - 48 + (4*n^3 + 20*n^2 + 28*n + 12)*x + (n^3 + 5*n^2 + 7*n + 3)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 8; for(n = 1, N, my(i = n + 1); v[i] = (-(16*n^3 + 112*n^2 + 256*n + 192 - (4*n^3 - 164*n^2 - 1280*n + 48)*x + (16*n^4 + 559*n^3 - 599*n^2 - 384*n - 12)*x^2 + (168*n^5 + 196*n^4 - 76*n^3 - 224*n^2 - 160*n)*x^3 + (16*n^5 + 28*n^4 - 9*n^3 - 29*n^2 - 6*n)*x^4)*v[i - 1] - ((32*n^2 + 256*n + 624)*x + (40*n^2 - 760*n - 1188)*x^2 - (200*n^3 + 1166*n^2 - 484*n - 456)*x^3 - (336*n^4 + 408*n^3 - 136*n^2 - 380*n - 228)*x^4 - (32*n^4 + 56*n^3 - 18*n^2 - 58*n - 12)*x^5)*deriv(v[i - 1], x) - ((48*n + 176)*x^2 - (16*n^2 + 468*n + 388)*x^3 - (168*n^3 + 348*n^2 - 145*n - 122)*x^4 + (152*n^3 + 320*n^2 + 240*n + 84)*x^5 + (16*n^3 + 44*n^2 + 35*n + 6)*x^6)*deriv(deriv(v[i - 1], x), x))/(-16*n^3 - 80*n^2 - 112*n - 48 + (4*n^3 + 20*n^2 + 28*n + 12)*x + (n^3 + 5*n^2 + 7*n + 3)*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-7a4827a398e28fe4","steps":["reverse","(\u03b8+1)\u00b2"],"edges":["e-8b10d4d15ed56acc","e-c94c8eae24ae549c"],"text":"A352880 \u2192 reverse \u2192 (\u03b8+1)\u00b2"},"headline":"A352880 \u2192 reverse \u2192 (\u03b8+1)\u00b2","chain":null}